Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 690341 7 pageshttpdxdoiorg1011552013690341

Research ArticleThe Viro Method for Construction of 119862119903 PiecewiseAlgebraic Hypersurfaces

Yisheng Lai1 Weiping Du1 and Renhong Wang2

1 Department of Information and Computing Science Zhejiang Gongshang University Hangzhou 310018 China2 Institute of Mathematical Sciences Dalian University of Technology Dalian 116024 China

Correspondence should be addressed to Yisheng Lai laiyisheng67163com

Received 30 July 2013 Accepted 22 August 2013

Academic Editor Lawrence Narici

Copyright copy 2013 Yisheng Lai et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We propose a newmethod to construct a real piecewise algebraic hypersurface of a given degree with a prescribed smoothness andtopology The method is based on the smooth blending theory and the Viro method for construction of Bernstein-Bezier algebraichypersurface piece on a simplex

1 Introduction

Let Δ be a simplicial subdivision of a region Ω in 119877119896 Δ iscalled a pure simplicial complex of dimension 119896 and canbe described as a finite collection of simplices such thatthe faces of each element of Δ are elements of Δ and theintersection of any two elements of Δ is an element of Δand every maximal element of Δ (with respect to inclusion)is a 119896-dimensional simplex We will sometimes refer to the119898-dimensional elements of Δ as 119898-cells and the simplicialsubdivision as a 119896-complex If two 119896-dimensional simplices inΔmeet in a face of dimension 119896minus1 we say they are adjacentΔis said to be hereditary if for every 120591 isin Δ (including the emptyset) any two 119899-dimensional simplices 119878 1198781015840 of Δ that contain 120591can be connected by a sequence 119878 = 1198781 1198782 119878119898 = 119878

1015840 in Δsuch that each 119878119894 is 119896-dimensional each 119878119894 contains 120591 and 119878119894and 119878119894+1 are adjacent for each 119894 (see [1 2])

Let Δ be a pure hereditary 119896-dimensional simplicialcomplex in 119877119896 let 1198781 1198782 119878119902 be a given fixed orderingof the 119896-cells in Δ and let Ω = ⋃

119902

119894=1119878119894 Now we recall the

definitions of 119862119903(Δ) and 119862119903

119899(Δ) (see [1 2])

Definition 1 For a nonnegative integer 119903 and a 119896-complex Δ119862119903(Δ) is the set of 119862119903 functions 119891 on Ω (ie functions such

that all 119903th order partial derivatives exist and are continuousonΩ) such that for every 120575 isin Δ including those of dimensionlt 119896 the restriction 119891|

120575is a polynomial function 119891|

120575isin

119877[1199091 119909119896] 119862119903

119899(Δ) is the subset of 119891 isin 119862119903

(Δ) such that therestriction of 119891 to each cell in Δ is a polynomial function ofdegree 119896 or less

It is clear that 119862119903(Δ) and 119862119903

119899(Δ) are a Noether ring and

a finite dimensional linear vector space respectively and arecalled a 119862119903 spline ring and a multivariate spline space withdegree 119899 and smoothness 119903 respectively We call

Z (119891) = (1199091 119909119896) isin Ω | 119891 (1199091 119909119896) = 0 (1)

a real 119862119903piecewise algebraic hypersurface (see [1ndash3]) where119891 isin 119862

119903(Δ)

An important direction in real algebraic geometry duringthe last three decades is the construction of real algebraichypersurfaces of a given degree with prescribed topologyCentral to these developments is a combinatorial construc-tion due to the Viro method [4ndash6] The Viro method is apowerful construction method of real nonsingular algebraichypersurfaces with prescribed topology (see [4ndash14]) It pro-vides a link between the topology of real algebraic varietiesand toric varieties It is based on polyhedral subdivisionsof the Newton polytopes A particular and important caseof the Viro method is called combinatorial patchworkingthe combinatorial patchworking is a particular case of theViro method which is characterized by the following twoproperties the subdivision used is a triangulation and each

2 Abstract and Applied Analysis

monomial of any ldquoblockrdquo polynomial corresponds to a vertexof the Newton simplex

Roughly speaking the Viro method starts with a convex(or coherent) polyhedral subdivision 119875119894 119894 isin 119868 of a polytope119875 and a collection 119891119894 119894 isin 119868 of real nondegenerate poly-nomials 119891119894 with Newton polyhedra 119875119894 whose truncations oncommon faces of Newton polyhedra coincide Then a Viropolynomial 119891 with Newton polytope 119875 is defined and Virorsquostheorem asserts that the topology of the real hypersurface119885(119891) defined by 119891 can be recovered by gluing together piecesof the real hypersurfaces 119885(119891119894)

It is well known that the hypersurface generally possessescomplex topological or geometric structures in CAGD andgeometric modelling Moreover the surface can be repre-sented in Bernstein-Bezier form since it is often defined ona simplex and writing a polynomial in itrsquos the Bernstein-Bezier representation has significant advantages since itscoefficients reflect geometric information about the shapeof the polynomial surface and the barycentric coordinatesrelative to the simplex are affine invariant and Bernstein-Bezier basis polynomials exhibit many important properties(see [3 15ndash17]) Therefore based on the Viro method andthe Newton polyhedra of Bernstein-Bezier polynomial Lai etal in [18] established a new method for the construction ofBernstein-Bezier algebraic hypersurfaces on a simplex with aprescribed topology and presented a method to describe thetopology of the Viro Bernstein-Bezier algebraic hypersurfacepiece

In CAGD and geometric modelling most of the complexcurves and surfaces are expressed by piecewise polynomialswith certain smoothness (see [3 17 18])Thus the aim of thispaper is to establish a new method for the construction ofreal piecewise algebraic hypersurfaces of a given degree withcertain smoothness and prescribed topology The new Viromethod is based on the work of Lai et al in [18] and thesmooth blending theory

The paper is organized as follows Section 2 reviewsbriefly the Viro method for the construction of Bernstein-Bezier algebraic hypersurface piece on a simplex In Section 3we define the chart of the piecewise polynomial and deal withsome properties of the chart of the Bernstein-Bezier poly-nomial and the chart of the piecewise polynomial Section 4is devoted to a new method for the construction of thereal piecewise algebraic hypersurfaces of a given degree withcertain smoothness and prescribed topology The method isprimarily based on the work of Lai et al in [18] and thesmooth blending theory

2 Construction of the Bernstein-BeacutezierAlgebraic Hypersurface Piece

This section reviews briefly the Viro method for the con-struction of Bernstein- Bezier algebraic hypersurface pieceas stated in [18]

Throughout this paper we denote by R+ (resp Rlowast

+) the

set of real numbers 119909 such that 119909 ge 0 (resp 119909 gt 0) and byZ+

the set of nonnegative integers Let 119878 = [V1 V119896+1] will bea fixed 119896-dimensional simplex with vertices V1 V119896+1

It is well known (cf [3 15]) that for any point 119901 isin 119878 it canbe expressed uniquely as

119901 =

119896+1

sum

119895=1

120591119895V119895 (2)

where sum119896+1

119895=1120591119895 = 1 120591119895 ge 0 119895 = 1 2 119896 + 1 and 120591 =

(1205911 120591119896+1) is the barycentric coordinates of 119901 with respectto 119878 (we abusively confuse a point 119901 = sum119896+1

119895=1120591119895V119895 isin 119878 with the

corresponding barycentric coordinates 120591 = (1205911 120591119896+1))Let

Z+(119896119899) = 120582 = (1205821 120582119896+1) isin (Z+)119896+1

| |120582| =

119896+1

sum

119894=1

120582119894 = 119899

(3)

We call a point 119901 isin 119878 domain point if 119901 =

(1205821119899 120582119896+1119899) with (1205821 120582119896+1) isin Z+(119896119899) Furtheron polyhedron relative to the simplex 119878 means a convexpolyhedron in 119878 with domain points as its vertices

It is well known (cf [3 15]) that for any polynomial119891over119878 with 119896 variables and degree at most 119899 it can be representedin the Bernstein-Bezier form as follows

119891 (120591) = sum

|120582|=119899

119887120582119861120582119899 (120591) (4)

where 119887120582 isin R 120582 = (1205821 120582119896+1) isin Z+(119896119899)

119861120582119899 (120591) =

119899

1205821 sdot sdot sdot 120582119896+1

1205911205821

1sdot sdot sdot 120591

120582119896+1

119896+1(5)

are the Bernstein basis of degree 119899 relative to 119878 The 119891(120591)is called a Bernstein-Bezier polynomial or B-form of thepolynomial 119891 relative to the simplex 119878 We will refer tothe polynomials in B-form as BB-polynomials and to theircoefficients 119887120582 as BB-coefficientsThe set of its zero points in 119878is called a Bernstein-Bezier algebraic hypersurface piece (BB-algebraic hypersurface piece for short)

For the BB-polynomial 119891 defined in (4) set

120583 (119891) = (

1205821

119899

120582119896+1

119899

) | 120582 isin Z+(119896119899) 119887120582 = 0 (6)

The convex hull conv(120583(119891)) of 120583(119891) on 119878 denoted by 119873(119891)is called Newton polyhedron (relative to 119878) of the BB-polynomial 119891

For a set Γ sub 119878 and a BB-polynomial 119891 = sum|120582|=119899

119887120582119861120582119899(120591)denote BB-polynomial sum

|120582|=119899120582isinΓ119887120582119861120582119899(120591) by 119891

Γ It is calledthe Γ-truncation of 119891

A BB-polynomial 119891 is called nondegenerate if 119891 and anytruncation 119891120575 on a proper face 120575 of 119873(119891) has a nonsingularzero set in (Clowast

)119896+1

LetJ119878 be a subset of the set of domain points in simplex 119878and 119875119878 = conv(J119878) Define the moment map associated withJ119878 relative to the simplex 119878 120601J

119878

Int(119878) rarr Int(119875119878) by

120601J119878(120591) =

sum(1198941119899119894

119896+1119899)isinJ

119878

1205911198941

1sdot sdot sdot 120591

119894119896+1

119896+1(1198941119899 119894119896+1119899)

sum(1198941119899119894

119896+1119899)isinJ

119878

1205911198941

1sdot sdot sdot 120591

119894119896+1

119896+1

(7)

Abstract and Applied Analysis 3

where Int(119875119878) is the complement in 119875119878 of the union of all itsproper faces

Definition 2 Let 119891 be a BB-polynomial with a Newtonpolyhedron 119875 Then the closure of 120601120583(119891)(120591 isin Int(119878) 119891(120591) =0) in 119875 is called chart 119862119905119878(119891) of 119891 where 120601120583(119891) Int(119878) rarrInt(119875) is themomentmap associated with 120583(119891) relative to thesimplex 119878

Let 119875 1198751 119875119898 sub 119878 be polyhedra with 119875 = ⋃119898

119894=1119875119894 and

Int(119875119894)⋂ Int(119875119895) = 0 for 119894 = 119895 Assume that ] 119875 rarr R isa continuous piecewise linear nonnegative convex functionsatisfying the following conditions

(1) all the restrictions ]|119875119894

are linear(2) if the restriction of ] to an open set is linear then this

set is contained in one of the 119875119894(3) ](119875⋂119871 (119896119899)) sub Z where 119871 (119896119899) is the set of the domain

points

Then this function ] with this property is said to convexify1198751 119875119898

Let1198911 119891119898 be BB-polynomials overR in 119896+1 variableswith119873(119891119894) = 119875119894 Let 119891

119875119894⋂119875119895

119894= 119891

119875119894⋂119875119895

119895for any 119894 119895 Then there

exists a unique BB-polynomial 119891 with 119873(119891) = 119875 = ⋃119898

119894=1119875119894

and 119891119875119894= 119891119894 for 119894 = 1 119898 If 119891(120591) = sum|120582|=119899

119887120582119861120582119899(120591) and ]is a function convexifying 1198751 P119898 we put

119891119905 (120591) = sum

|120582|=119899

119887120582119861120582119899 (120591) 119905](120582119899)

(8)

The BB-polynomials 119891119905 are said to be obtained by119901119886119905119888ℎ119908119900119903119896119894119899119892 BB-polynomials 1198911 119891119898 by ] or briefly 119891119905is a 119901119886119905119888ℎ119908119900119903119896 of BB-polynomials 1198911 119891119898 by ]

Let 1198911 119891119898 be BB-polynomials in 119896 + 1 variables with119873(119891119894) = 119875119894 and let Int(119875119894)⋂ Int(119875119895) = 0 for 119894 = 119895 Achart 119862119905119878(119891) of a BB-polynomial 119891 with 119873(119891) = 119875 is saidto be obtained by 119901119886119905119888ℎ119908119900119903119896119894119899119892 charts of BB-polynomials1198911 119891119898 and it is a 119901119886119905119888ℎ119908119900119903119896 of charts of BB-polynomials1198911 119891119898 if 119875 = ⋃

119898

119894=1119875119894 and the chart 119862119905119878(119891) of 119891 up to

isotopy is⋃119898

119894=1119862119905119878(119891119894)

The result about the Viro method for the constructionof the Bernstein-Bezier algebraic hypersurface piece on asimplex with a prescribed topology is shown in the followingproposition (see [18])

Proposition 3 (see [18]) Let 119875 1198751 119875119898 ] 1198911 119891119898 and119891119905 be as above (119891119905 is a119901119886119905119888ℎ119908119900119903119896 of BB-polynomials1198911 119891119898by ]) If BB-polynomials 1198911 119891119898 are nondegenerate thenthere exists 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart ofBB-polynomial 119891119905 is obtained by patchworking charts of BB-polynomials 1198911 119891119898

3 The Chart of the Piecewise Polynomial

In this section the chart of the piecewise polynomial isdefined and some properties of the chart of the Bernstein-Bezier polynomial and the chart of the piecewise polynomialare discussed

Theorem 4 Let 119891 be a BB-polynomial with Newton polyhe-dron 119875 and let Γ be a face of 119875 Then

(1) 119862119905119878(119891)⋂ Γ = 119862119905119878(119891Γ)

(2) if 119891Γ is nondegenerate with respect to Γ (which is thecase when for example119891 is nondegenerate with respectto 119875) then the 119862119905119878(119891) intersects Γ transversally

Proof Suppose that the BB-polynomial 119891 is defined in (4)and

119892 (119909) = sum

|120582|=119899

119899

1205821 sdot sdot sdot 120582119896+1

1198871205821199091205822

2sdot sdot sdot 119909

120582119896+1

119896+1 (9)

Define the mapping 120595 119878 rarr 119879(119896119899) = (1199092 119909119896+1) isin

(R+)119896|1199092 + sdot sdot sdot + 119909119896+1 le 119899 by

120595 (1205911 1205912 120591119896+1) = (1198991205912 119899120591119896+1)

(1205911 1205912 120591119896+1) isin 119878

(10)

According to [18 Lemma 32] then 120595(119875) is the Newtonpolyhedron 119873(119892) of the polynomial 119892 and 120595(Γ) is a face of119873(119892) if Γ is a face of 119875 Moreover by [10 Remark 11] the119888ℎ119886119903119905 119862119905+(119892) of 119892 (see [6ndash8 18]) and the truncation 119892120595(Γ) onthe face 120595(Γ) have the following properties

(a) 119862119905+(119892)⋂120595(Γ) = 119862119905+(119892120595(Γ))

(b) the 119888ℎ119886119903119905 119862119905+(119892) intersects 120595(Γ) transversally if 119892120595(Γ)

is nondegenerate with respect to 120595(Γ)

On the other hand by [18 Lemma 32 andTheorem 35]we can get the following equalities

120595minus1(119862119905+ (119892)⋂120595 (Γ)) = 119862119905119878 (119891)⋂Γ

120595minus1(119862119905+ (119892

120595(Γ))) = 119862119905119878 (119891

Γ)

(11)

This together with equality (10) and properties (a) and (b)shows that 119862119905119878(119891)⋂ Γ = 119862119905119878(119891

Γ) and the 119862119905119878(119891) intersects

Γ transversally if 119891Γ is nondegenerate with respect to Γ Thiscompletes the proof

The following result is a generalization of Farinrsquos theorem(see [3 15]) on high dimensional space

Proposition 5 (see [3 15]) Let

119901(1)(120591) = sum

|120582|=119899

119887(1)

120582119861120582119899 (120591) 119901

(2)(120591) = sum

|120582|=119899

119887(2)

120582119861120582119899 (120591)

(12)

be BB-polynomials of degree 119899 that are defined on two adjacent119896-dimensional simplices 1198781 = [V1 V2 V119896+1] and 1198782 =

[V10158401 V2 V119896+1] respectively Then 119901(1)(120591) and 119901(2)(120591) are 119862119903

smoothly connected on [V2 V119896+1] if and only if for all 120588 isin0 1 119903 and 1205822 120582119896+1 isin Z+ with sum

119896+1

119895=2120582119895 = 119899 minus 120588

119887(2)

120582120588= sum

|120598|=120588

119887(1)

120598+1205820119861120598120588 (120591

1015840) (13)

4 Abstract and Applied Analysis

where 1205820 = (0 1205822 120582119896+1) 120582120588 = (120588 1205822 120582119896+1) and 120598 =(1205981 1205982 120598119896+1) Here 1205911015840 denotes the barycentric coordinate ofV10158401with respect to the simplex 1198781

Theorem 6 Suppose that 119901(1)(120591) and 119901(2)(120591) defined as aboveare 119862119903

(119903 ge 0) smoothly connected on 1198781⋂1198782 = [V2 V119896+1]Then

(1) 119873(119901(1))⋂(1198781⋂1198782) = 119873(119901(2))⋂(1198781⋂1198782) and Γ =

119873(119901(1))⋂(1198781⋂1198782) is a face of 119873(119901(1)) and 119873(119901(2))

if 119873(119901(1))⋂(1198781⋂1198782) is nonempty where 119873(119901(1)) and119873(119901

(2)) are Newton polyhedra of BB-polynomials 119901(1)

and 119901(2) respectively(2) 119862119905119878

1

(119901(1))⋂ Γ = 119862119905119878

2

(119901(2))⋂ Γ for Γ above

Proof By assumption and taking 120588 = 0 in equality (13) wehave that 119887(1)

1205820= 119887

(2)

1205820for all 1205822 120582119896+1 isin Z+ with sum119896+1

119895=2120582119895 =

119899 Thus 120583(119901(1))⋂(1198781⋂1198782) = 120583(119901(2))⋂(1198781⋂1198782) where 120583

is defined in (6) This implies that 119873(119901(1))⋂(1198781⋂1198782) =

119873(119901(2))⋂(1198781⋂1198782) and so Γ = 119873(119901(1))⋂(1198781⋂1198782) is a face

of119873(119901(1)) and119873(119901(2)) if119873(119901(1))⋂(1198781⋂1198782) is nonemptyAccording to the argument above it is easy to see that Γ-

truncations (119901(1))Γ (119901(2))Γ of 119901(1) and 119901(2) satisfy the equality(119901

(1))Γ= (119901

(2))Γ Thus we can get that

1198621199051198781

((119901(1))

Γ

) = 1198621199051198782

((119901(2))

Γ

) (14)

by the definition of the 119888ℎ119886119903119905On the other hand it follows fromTheorem 4 that

1198621199051198781

(119901(1))⋂ Γ = 119862119905119878

1

((119901(1))

Γ

)

1198621199051198782

(119901(2))⋂ Γ = 119862119905119878

2

((119901(2))

Γ

)

(15)

Therefore

1198621199051198781

(119901(1))⋂Γ = 119862119905119878

2

(119901(2))⋂Γ (16)

by equality (14) This completes the proof

LetΔ be a pure hereditary simplicial complexwith 119896-cells(ie 119896-dimensional simplex) 1198781 119878119902 Let 119891 isin 119862

119903

119899(Δ) and

each polynomial 119891|119878120572

be expressed in the B-form

1198911003816100381610038161003816119878120572

(120591) = sum

|120582|=119899

119887[120572]

120582119861120582119899 (120591) 120572 = 1 2 119902 (17)

The piecewise polynomial 119891 defined as above is called 119862119903

piecewise BB-polynomial

Theorem 7 Let 119891 isin 119862119903

119899(Δ) be a piecewise BB-polynomial

defined in (17) Then for each adjacent pair 119878120572 119878120573 of 119896-cellsin Δ

(1) 119873(119891|119878120572

)⋂(119878120572⋂119878120573) = 119873(119891|119878120573

)⋂(119878120572⋂119878120573) andΓ120572120573 = 119873(119891|119878

120572

)⋂(119878120572⋂119878120573) is a face of 119873(119891|119878120572

) and119873(119891|119878

120573

) if 119873(119891|119878120572

)⋂(119878120572⋂119878120573) is nonempty where

119873(119891|119878120572

) and 119873(119891|119878120573

) are Newton polyhedra of BB-polynomials 119891|119878

120572

and 119891|119878120573

respectively

(2) 119862119905119878120572

(119891|119878120572

)⋂ Γ120572120573 = 119862119905119878120573

(119891|119878120573

)⋂ Γ120572120573 for Γ120572120573 above

Proof We can get the conclusion from Theorem 6 immedi-ately

According to Theorems 4ndash7 we can define the 119888ℎ119886119903119905 of apiecewise BB-polynomial

Definition 8 Let 119891 isin 119862119903

119899(Δ) be a piecewise BB-polynomial

defined in (17) and let 119873(119891|119878120572

) be the Newton polyhedronof the BB-polynomial 119891|119878

120572

The chart 119862119905(119891) of 119891 is theclosure of ⋃119902

120572=1120601120583(119891|

119878120572)(120591 isin Int(119878120572) 119891|119878

120572

(120591) = 0) where120583(119891|119878

120572

) = (1205821119899 120582119896+1119899)|120582 isin Z+(119896119899) 119887[120572]

120582= 0 and

120601120583(119891|119878120572) Int(119878120572) rarr Int(119873(119891|119878

120572

)) is the moment mapassociated with 120583(119891|119878

120572

) relative to the simplex 119878120572

Obviously we can get the following conclusion fromTheorems 4ndash7 and the definition above immediately

Theorem 9 Let 119891 isin 119862119903

119899(Δ) be a piecewise BB-polynomial

defined in (17) Then the 119888ℎ119886119903119905 119862119905(119891) of 119891 is⋃119902

120572=1119862119905119878120572

(119891|119878120572

)

4 The Construction of Real 119862119903 PiecewiseAlgebraic Hypersurfaces

In this section we propose a new method for the construc-tion of real piecewise algebraic hypersurfaces of a givendegree with certain smoothness and prescribed topologyThemethod is primarily based on the work of Lai et al in [18] andthe smooth blending theory

LetΔ be a pure hereditary simplicial complexwith 119896-cells(ie 119896-dimensional simplex) 1198781 119878119902

For each adjacent pair 119878120572 = [V(1205721) V(1205722) V(120572119896+1)]119878120573 = [V(1205731) V(1205732) V(120573119896+1)] of 119896-cells in Δ assume that119878120572 cap 119878120573 = [V(1205721) V(120572119894minus1) V(120572119894) V(120572119894+1) V(120572119896+1)] =

[V(1205731) V(120573119895minus1) V(120573119895) V(120573119895+1) V(120573119896+1)] where the hatmeans that the corresponding vertex is omitted Obvi-ously there exist two arrangements 1198971 1198972 119897119896+1 and 1198891 1198892 119889119896+1 of numbers 1 2 119896 + 1 such that V(120572119897

1) =

V(120572119894) V(1205731198891) = V(120573119895) and V(120572119897

119908) = V(120573119889

119908) for 119908 = 2 119896 + 1

Now we define two one-to-one transformations 120578120572120573120578120573120572 (R+)

119896+1rarr (R+)

119896+1 by 120578120572120573(120582) = (1205821198971

1205821198972

120582119897119896+1

)and 120578120573120572(120582) = (120582119889

1

1205821198892

120582119889119896+1

) respectively where 120582 =

(1205821 1205822 120582119896+1) isin (R+)119896+1

For any given 120572 isin 1 119902 let 119891(1205721) 119891

(120572119898120572) be BB-

polynomials over the simplex 119878120572 = [V(1205721) V(1205722) V(120572119896+1)]with119873(119891(120572120577)

) = 119875(120572120577) 120577 = 1 119898120572 and let

119891[120572]

119905(120591) = sum

|120582|=119899

119887[120572]

120582119861120582119899 (120591) 119905

][120572](120582119899)(18)

be a 119901119886119905119888ℎ119908119900119903119896119894119899119892 of BB-polynomials 119891(1205721) 119891

(120572119898120572) by

][120572]

Abstract and Applied Analysis 5

Define the piecewise BB-polynomial 119891119905 on the domainΩby

119891119905

1003816100381610038161003816119878120572

(120591) = 119891[120572]

119905(120591) 120572 = 1 2 119902 (19)

Theorem10 Suppose that119891(11) 119891

(11198981) 119891

(21) 119891

(119902119898119902)

are non-degenerate that 119891[120572]

119905is a 119901119886119905119888ℎ119908119900119903119896119894119899119892 of BB-

polynomials 119891(1205721) 119891

(120572119898120572) by ][120572] 120572 = 1 119902 and that

119878120572 119878120573 V(120572119894) V(120573119895) 120578120572120573 120578120573120572 119875(120572120577) 119891[120572]

119905 and 119891119905 are defined

as above and Int(119875(120572120577))⋂ Int(119875(120572120589)) = 0 for 120577 = 120589 If for eachadjacent pair 119878120572 119878120573 of 119896-cells in Δ and all 120588 isin 0 1 119903 and120582 = (1205821 1205822 120582119896+1) isin Z+(119896119899) with sum

119896+1

119894=2120582119894 = 119899 minus 120588

119887[120573]

(120578minus1

120573120572(120582120588))119905][120573](120578minus1

120573120572(120582120588)119899)

= sum

|120598|=120588

119887[120572]

120578minus1

120572120573(120598+1205820)119861120598120588 (120591

1015840

120573120572) 119905

][120572](120578minus1120572120573(120598+1205820)119899) 119905 isin R

lowast

+

(20)

where 120582120588 = (120588 1205822 120582119896+1) and 1205911015840120573120572 denotes the barycen-tric coordinate of V(120573119895) with respect to the simplex 119878120572 =

[V(1205721198971) V(120572119897

2) V(120572119897

119896+1)] then

(1) 119891119905 isin 119862119903

119899(Δ)

(2) if for any BB-polynomial 119891(120572119894) with 119873(119891(120572119894))⋂

(119878120572⋂119878120573) = 0 119894 isin 1 119898120572 there exists a BB-polynomial 119891(120573119895) in 119891

(1205731) 119891

(120573119898120573) such that

119873(119891(120573119895)

)⋂(119878120572⋂119878120573) = 119873(119891(120572119894))⋂(119878120572⋂119878120573) then

there is 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart119862119905(119891119905) of piecewise BB-polynomial 119891119905 up to isotopy is⋃

119902

120572=1⋃

119898120572

119894=1119862119905119878120572

(119891(120572119894))

Proof Let 120591 = (1205911 120591119896+1) and 120594 = (1205941 1205942 120594119896+1) bethe barycentric coordinates of point 119901 isin 119878120573 with respectto 119878120573 = [V(1205731) V(1205732) V(120573119896+1)] and 119878120573 = [V(120573119889

1) V(120573119889

2)

V(120573119889119896+1

)] respectively Then

120594 = (1205941 1205942 120594119896+1) = 120578120573120572 (120591) = (1205911198891

1205911198892

120591119889119896+1

) (21)

by the definition of the transformation 120578120573120572 and so

119891[120573]

119905(119901) = 119891

[120573]

119905 (120591)

= sum

|120582|=119899

119887[120573]

120582

119899

12058211205822 sdot sdot sdot 120582119896+1

times 1205911205821

11205911205822

2sdot sdot sdot 120591

120582119896+1

119896+1119905][120573](120582119899)

= sum

|120582|=119899

119887[120573]

120582

119899

1205821198891

1205821198891

sdot sdot sdot 120582119889119896+1

times (1205941)1205821198891(1205942)

1205821198892sdot sdot sdot (120594119896+1)

120582119889119896+1 119905

][120573](120582119899)

(22)

Set 120583 = (1205831 1205832 120583119896+1) = 120578120573120572(120582) Then 120582 = 120578minus1120573120572(120583) and

(1205831 1205832 120583119896+1) = (1205821198891

1205821198892

120582119889119896+1

) Therefore it followsfrom equality (22) that

119891[120573]

119905(119901) = sum

|120583|=119899

119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899)

times

119899

12058311205831 sdot sdot sdot 120583119896+1

(1205941)1205831

(1205942)1205832

sdot sdot sdot (120594119896+1)120583119896+1

(23)

This shows that the polynomial 119891[120573]

119905over 119878120573 = [V(120573119889

1) V(120573119889

2)

V(120573119889119896+1

)] can be represented as follows

119891[120573]

119905(120594) = sum

|120583|=119899

119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899)

119861120583119899 (120594) (24)

By a similar argument above we have that the polynomial119891[120572]

119905over 119878120572 = [V(120572119897

1) V(120572119897

119896+1)] can be represented as

follows

119891[120572]

119905(120594) = sum

|120583|=119899

119887[120572]

120578minus1

120572120573(120583)119905][120572](120578minus1

120572120573(120583)119899)

119861120583119899 (120594) (25)

where 120594 = (1205941 120594

2 120594

119896+1) is the barycentric coordinate of

point 119901 isin 119878120572 with respect to 119878120572 = [V(1205721198971) V(120572119897

2) V(120572119897

119896+1)]

Since V(120572119897119908) = V(120573119889

119908) for 119908 = 2 119896 + 1 119878120572⋂119878120573 =

[V(1205721198972) V(120572119897

119896+1)] thus according to Proposition 5 119891119905 isin

119862119903

119899(Δ) if and only if the BB-coefficients 119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899) of

119891[120573]

119905and the BB-coefficients 119887[120572]

120578minus1

120572120573(120583)119905][120572](120578minus1

120572120573(120583)119899) of 119891[120572]

119905satisfy

equality (20) This proves property (1)Below we will show that property (2) holdsFor each adjacent pair 119878120572 119878120573 of 119896-cells in Δ set Γ120572120573 =

119873(119891(120572119894))⋂(119878120572⋂119878120573)when119873(119891

(120572119894))⋂(119878120572⋂119878120573) = 0 then Γ120572120573

is a face of 119873(119891(120572119894)) and 119873(119891(120573119895)

) by the assumption that119873(119891

(120573119895))⋂(119878120572⋂119878120573) = 119873(119891

(120572119894))⋂(119878120572⋂119878120573) Since 119891[120572]

119905

(resp 119891[120573]

119905) is a patchworking of BB-polynomials 119891(1205721)

119891(120572119898120572) (resp 119891(1205731)

119891(120573119898120573)) by ][120572] (resp ][120573]) and

Int(119875(120572120577))⋂ Int(119875(120572120589)) = Int(119875(120573120577))⋂ Int(119875(120573120589)) = 0 for 120577 = 120589then for Γ120572120573-truncations (119891

(120572119894))Γ120572120573 (119891(120573119895)

)Γ120572120573 (119891[120572]

119905)Γ120572120573 and

(119891[120573]

119905)Γ120572120573 of 119891(120572119894) 119891(120573119895) 119891[120572]

119905 and 119891[120573]

119905 respectively we have

that

(119891(120572119894))

Γ120572120573

= (119891[120572]

119905)

Γ1205721205731003816100381610038161003816100381610038161003816119905=1

(119891(120573119895)

)

Γ120572120573

= (119891[120573]

119905)

Γ1205721205731003816100381610038161003816100381610038161003816119905=1

(26)

Since119891[120572]

119905and119891[120573]

119905are119862119903 smoothly connected on 119878120572⋂119878120573

119873(119891(120573119895)

)⋂(119878120572⋂119878120573) = 119873(119891(120572119894))⋂(119878120572⋂119878120573) and (20) implies

that (119891[120572]

119905)Γ120572120573= (119891

[120573]

119905)Γ120572120573 Thus (119891(120572119894)

)Γ120572120573= (119891

(120573119895))Γ120572120573 by (26)

6 Abstract and Applied Analysis

Using a similar argument in the proof of Theorem 6 wehave 119862119905119878

120572

(119891(120572119894))⋂ Γ120572120573 = 119862119905119878

120573

(119891(120573119895)

)⋂ Γ120572120573 by (119891(120572119894))Γ120572120573=

(119891(120573119895)

)Γ120572120573 This shows that

119898120572

⋃

119894=1

119862119905119878120572

(119891(120572119894))⋂(119878120572⋂119878120573)

=

119898120573

⋃

119895=1

119862119905119878120573

(119891(120573119895)

)⋂(119878120572⋂119878120573)

(27)

On the other hand since 119891[120572]

119905is a patchworking of

BB-polynomials 119891(1205721) 119891

(120572119898120572) by ][120572] we get from the

assumption and Proposition 3 that there is 119905[120572]0gt 0 such that

for any 119905 isin (0 119905[120572]0] the chart119862119905119878

120572

(119891[120572]

119905) of BB-polynomial119891[120572]

119905

(ie 119891119905|119878120572

) up to isotopy is⋃119898120572

119894=1119862119905119878120572

(119891(120572119894)) This conclusion

together with Theorems 7ndash9 and equality (27) implies thatthere is 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart119862119905(119891119905) of piecewise BB-polynomial 119891119905 up to isotopy is⋃

119902

120572=1⋃

119898120572

119894=1119862119905119878120572

(119891(120572119894)) where 1199050 = min119905[1]

0 119905

[119902]

0 This

completes the proof

Assume that 119875 is a 119896-dimensional simplex with domainpoints in 119878 as its vertices and that the BB-polynomial 119891 is areal 119896 + 1-nomial (this means that the nonzero coefficients in119891 correspond to the only vertices of 119875) Denote by119872(119875) theset of middle points of edges of 119875 For any point V isin 119872(119875) onan edge with endpoints (1198941119899 119894119896+1119899) (1198951119899 119895119896+1119899)we assign that

120575 (V) = sign (119887(1198941119894119896+1

) sdot 119887(1198951119895119896+1

)) (28)

where 119887(1198941119894119896+1

) 119887(1198951119895119896+1

) are the corresponding BB-coefficients in 119891 Put

119872minus (119875) = V isin 119872 (119875) | 120575 (V) = minus1 (29)

Theorem 11 Suppose that 119891(11) 119891

(11198981) 119891

(1199021)

119891(119902119898119902) 119875(11) 119875(1119898

1) 119875(1199021) 119875(119902119898

119902) 119891

[120572]

119905 and119891119905 sat-

isfy the conditions in Theorem 10 and that for each 120572 isin

1 2 119902 119891(1205721) 119891

(120572119898120572) are real (119896 + 1)-nomial and

119875(1205721) 119875(120572119898120572) with dimension 119896 form a triangulation of

119873(119891[120572]

119905) Then 119891119905 isin 119862119903

119899(Δ) and there exists a constant 1199050 gt 0

such that for any 119905 isin (0 1199050] the chart 119862119905(119891) of 119891 up to isotopyis ⋃119902

120572=1⋃

119898120572

119894=1conv(119872minus(119875(120572119894))) where 119872minus(119875(120572119894)) is defined as

(29)

Proof By assumption and [18 Theorem 44] we know thatthere is 119905[120572]

0gt 0 such that for any 119905 isin (0 119905

[120572]

0] the

chart 119862119905119878120572

(119891[120572]

119905) of BB-polynomial 119891[120572]

119905 up to isotopy is

⋃119898120572

119894=1conv(119872minus(119875(120572119894)))

The remainder of the proof can be completed by a similarapproach in the proof of Theorem 10

According toTheorems 10 and 11 if we want to constructa 119862119903 piecewise algebraic hypersurface 119891(120591) = 0 with aprescribed complex topology and a degree on a partition

Δ we can just construct some Bernstein-Bezier algebraichypersurfaces pieces 119891(1205721)

(120591) = 0 119891(120572119898120572)(120591) = 0 with

simple topology and a convex function ][120572] on each 119896-cell119878120572 where 119891

(1205721) 119891

(120572119898120572) and ][120572] satisfy the conditions

of Theorem 10 or Theorem 11 120572 = 1 2 119902 and then theBB-polynomial 119891[120572]

(120591) is obtained by a patchworking of BB-polynomials 119891(1205721)

119891(120572119898120572) by ][120572] Thus we can get 119891(120591)

by defining 119891|119878120572

(120591) = 119891[120572](120591)

Conflict of Interests

The authors do not have any possible conflict of interests inthis paper

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (Grant nos 11271328 11071031U0935004 U1135003 11290143 11101366 11226329 and61272307) the Zhejiang Provincial Natural Science Foun-dation (nos Y7080068 LQ13A010004) and the Foundationof Department of Education of Zhejiang Province (noY201223556)

References

[1] Y S Lai R H Wang and J M Wu ldquoReal zeros of the zero-dimensional parametric piecewise algebraic varietyrdquo Science inChina A vol 52 no 4 pp 817ndash832 2009

[2] Y S Lai R H Wang and J M Wu ldquoSolving parametricpiecewise polynomial systemsrdquo Journal of Computational andApplied Mathematics vol 236 no 5 pp 924ndash936 2011

[3] R-H Wang X Q Shi Z X Luo and Z X Su Multivariatespline and its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001

[4] O Ya Viro ldquoGluing of algebraic hypersurfaces smoothingof singularities and constructionof curvesrdquo in Proceedings ofthe Leningrad International Topological Conference pp 149ndash197Nauka Leningrad Russia 1983 (Russian)

[5] O Ya Viro ldquoGluing of plane real algebraic curves and construc-tions of curves of degrees 6 and 7rdquo inTopology (Leningrad 1982)vol 1060 of Lecture Notes inMathematics pp 187ndash200 SpringerBerlin Germany 1984

[6] O Ya Viro ldquoPatchworking real algebraic varietiesrdquo httpwwwmathsunysbedusimolegpwpdf

[7] E Shustin ldquoCritical points of real polynomials subdivisionsof Newton polyhedra and topology of real algebraic hypersur-facesrdquo Transactions of the American Mathematical Society vol173 no 2 pp 203ndash223 1996

[8] E Shustin ldquoGluing of singular and critical pointsrdquoTopology vol37 no 1 pp 195ndash217 1998

[9] E Shustin ldquoReal plane algebraic curves with prescribed singu-laritiesrdquo Topology vol 32 no 4 pp 845ndash856 1993

[10] F Bihan ldquoViro method for the construction of real completeintersectionsrdquo Advances in Mathematics vol 169 no 2 pp 177ndash186 2002

[11] J J Risler ldquoConstruction drsquohypersurfaces reelles [drsquoapres Viro]rdquoin Seminaire Bourbaki vol 199293 of Asterique no 216 Exp7633 pp 69ndash86 1993

Abstract and Applied Analysis 7

[12] I Itenberg and E Shustin ldquoSingular points and limit cycles ofplanar polynomial vector fieldsrdquo Duke Mathematical Journalvol 102 no 1 pp 1ndash37 2000

[13] I Itenberg and E Shustin ldquoViro theorem and topology of realand complex combinatorial hypersurfacesrdquo Israel Journal ofMathematics vol 133 no 1 pp 189ndash238 2003

[14] B Sturmfels ldquoVirorsquos theorem for complete intersectionsrdquoAnnalidella Scuola Normale Superiore di Pisa vol 21 no 3 pp 377ndash386 1994

[15] G Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Guide (Computer Science and ScientificComputing) Academic Press San Diego Calif USA 4thedition 1997

[16] C M Hoffmann Geometric and Solid Modeling Morgan Kauf-mann San Mateo Calif USA 1989

[17] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design AK Peters Boston Mass USA 1993

[18] Y S Lai W P Du and R H Wang ldquoThe Viro method forconstruction of Bernstein-Bezier algebraic hypersurface piecerdquoScience China Mathematics vol 55 no 6 pp 1269ndash1279 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

monomial of any ldquoblockrdquo polynomial corresponds to a vertexof the Newton simplex

Roughly speaking the Viro method starts with a convex(or coherent) polyhedral subdivision 119875119894 119894 isin 119868 of a polytope119875 and a collection 119891119894 119894 isin 119868 of real nondegenerate poly-nomials 119891119894 with Newton polyhedra 119875119894 whose truncations oncommon faces of Newton polyhedra coincide Then a Viropolynomial 119891 with Newton polytope 119875 is defined and Virorsquostheorem asserts that the topology of the real hypersurface119885(119891) defined by 119891 can be recovered by gluing together piecesof the real hypersurfaces 119885(119891119894)

It is well known that the hypersurface generally possessescomplex topological or geometric structures in CAGD andgeometric modelling Moreover the surface can be repre-sented in Bernstein-Bezier form since it is often defined ona simplex and writing a polynomial in itrsquos the Bernstein-Bezier representation has significant advantages since itscoefficients reflect geometric information about the shapeof the polynomial surface and the barycentric coordinatesrelative to the simplex are affine invariant and Bernstein-Bezier basis polynomials exhibit many important properties(see [3 15ndash17]) Therefore based on the Viro method andthe Newton polyhedra of Bernstein-Bezier polynomial Lai etal in [18] established a new method for the construction ofBernstein-Bezier algebraic hypersurfaces on a simplex with aprescribed topology and presented a method to describe thetopology of the Viro Bernstein-Bezier algebraic hypersurfacepiece

In CAGD and geometric modelling most of the complexcurves and surfaces are expressed by piecewise polynomialswith certain smoothness (see [3 17 18])Thus the aim of thispaper is to establish a new method for the construction ofreal piecewise algebraic hypersurfaces of a given degree withcertain smoothness and prescribed topology The new Viromethod is based on the work of Lai et al in [18] and thesmooth blending theory

The paper is organized as follows Section 2 reviewsbriefly the Viro method for the construction of Bernstein-Bezier algebraic hypersurface piece on a simplex In Section 3we define the chart of the piecewise polynomial and deal withsome properties of the chart of the Bernstein-Bezier poly-nomial and the chart of the piecewise polynomial Section 4is devoted to a new method for the construction of thereal piecewise algebraic hypersurfaces of a given degree withcertain smoothness and prescribed topology The method isprimarily based on the work of Lai et al in [18] and thesmooth blending theory

2 Construction of the Bernstein-BeacutezierAlgebraic Hypersurface Piece

This section reviews briefly the Viro method for the con-struction of Bernstein- Bezier algebraic hypersurface pieceas stated in [18]

Throughout this paper we denote by R+ (resp Rlowast

+) the

set of real numbers 119909 such that 119909 ge 0 (resp 119909 gt 0) and byZ+

the set of nonnegative integers Let 119878 = [V1 V119896+1] will bea fixed 119896-dimensional simplex with vertices V1 V119896+1

It is well known (cf [3 15]) that for any point 119901 isin 119878 it canbe expressed uniquely as

119901 =

119896+1

sum

119895=1

120591119895V119895 (2)

where sum119896+1

119895=1120591119895 = 1 120591119895 ge 0 119895 = 1 2 119896 + 1 and 120591 =

(1205911 120591119896+1) is the barycentric coordinates of 119901 with respectto 119878 (we abusively confuse a point 119901 = sum119896+1

119895=1120591119895V119895 isin 119878 with the

corresponding barycentric coordinates 120591 = (1205911 120591119896+1))Let

Z+(119896119899) = 120582 = (1205821 120582119896+1) isin (Z+)119896+1

| |120582| =

119896+1

sum

119894=1

120582119894 = 119899

(3)

We call a point 119901 isin 119878 domain point if 119901 =

(1205821119899 120582119896+1119899) with (1205821 120582119896+1) isin Z+(119896119899) Furtheron polyhedron relative to the simplex 119878 means a convexpolyhedron in 119878 with domain points as its vertices

It is well known (cf [3 15]) that for any polynomial119891over119878 with 119896 variables and degree at most 119899 it can be representedin the Bernstein-Bezier form as follows

119891 (120591) = sum

|120582|=119899

119887120582119861120582119899 (120591) (4)

where 119887120582 isin R 120582 = (1205821 120582119896+1) isin Z+(119896119899)

119861120582119899 (120591) =

119899

1205821 sdot sdot sdot 120582119896+1

1205911205821

1sdot sdot sdot 120591

120582119896+1

119896+1(5)

are the Bernstein basis of degree 119899 relative to 119878 The 119891(120591)is called a Bernstein-Bezier polynomial or B-form of thepolynomial 119891 relative to the simplex 119878 We will refer tothe polynomials in B-form as BB-polynomials and to theircoefficients 119887120582 as BB-coefficientsThe set of its zero points in 119878is called a Bernstein-Bezier algebraic hypersurface piece (BB-algebraic hypersurface piece for short)

For the BB-polynomial 119891 defined in (4) set

120583 (119891) = (

1205821

119899

120582119896+1

119899

) | 120582 isin Z+(119896119899) 119887120582 = 0 (6)

The convex hull conv(120583(119891)) of 120583(119891) on 119878 denoted by 119873(119891)is called Newton polyhedron (relative to 119878) of the BB-polynomial 119891

For a set Γ sub 119878 and a BB-polynomial 119891 = sum|120582|=119899

119887120582119861120582119899(120591)denote BB-polynomial sum

|120582|=119899120582isinΓ119887120582119861120582119899(120591) by 119891

Γ It is calledthe Γ-truncation of 119891

A BB-polynomial 119891 is called nondegenerate if 119891 and anytruncation 119891120575 on a proper face 120575 of 119873(119891) has a nonsingularzero set in (Clowast

)119896+1

LetJ119878 be a subset of the set of domain points in simplex 119878and 119875119878 = conv(J119878) Define the moment map associated withJ119878 relative to the simplex 119878 120601J

119878

Int(119878) rarr Int(119875119878) by

120601J119878(120591) =

sum(1198941119899119894

119896+1119899)isinJ

119878

1205911198941

1sdot sdot sdot 120591

119894119896+1

119896+1(1198941119899 119894119896+1119899)

sum(1198941119899119894

119896+1119899)isinJ

119878

1205911198941

1sdot sdot sdot 120591

119894119896+1

119896+1

(7)

Abstract and Applied Analysis 3

where Int(119875119878) is the complement in 119875119878 of the union of all itsproper faces

Definition 2 Let 119891 be a BB-polynomial with a Newtonpolyhedron 119875 Then the closure of 120601120583(119891)(120591 isin Int(119878) 119891(120591) =0) in 119875 is called chart 119862119905119878(119891) of 119891 where 120601120583(119891) Int(119878) rarrInt(119875) is themomentmap associated with 120583(119891) relative to thesimplex 119878

Let 119875 1198751 119875119898 sub 119878 be polyhedra with 119875 = ⋃119898

119894=1119875119894 and

Int(119875119894)⋂ Int(119875119895) = 0 for 119894 = 119895 Assume that ] 119875 rarr R isa continuous piecewise linear nonnegative convex functionsatisfying the following conditions

(1) all the restrictions ]|119875119894

are linear(2) if the restriction of ] to an open set is linear then this

set is contained in one of the 119875119894(3) ](119875⋂119871 (119896119899)) sub Z where 119871 (119896119899) is the set of the domain

points

Then this function ] with this property is said to convexify1198751 119875119898

Let1198911 119891119898 be BB-polynomials overR in 119896+1 variableswith119873(119891119894) = 119875119894 Let 119891

119875119894⋂119875119895

119894= 119891

119875119894⋂119875119895

119895for any 119894 119895 Then there

exists a unique BB-polynomial 119891 with 119873(119891) = 119875 = ⋃119898

119894=1119875119894

and 119891119875119894= 119891119894 for 119894 = 1 119898 If 119891(120591) = sum|120582|=119899

119887120582119861120582119899(120591) and ]is a function convexifying 1198751 P119898 we put

119891119905 (120591) = sum

|120582|=119899

119887120582119861120582119899 (120591) 119905](120582119899)

(8)

The BB-polynomials 119891119905 are said to be obtained by119901119886119905119888ℎ119908119900119903119896119894119899119892 BB-polynomials 1198911 119891119898 by ] or briefly 119891119905is a 119901119886119905119888ℎ119908119900119903119896 of BB-polynomials 1198911 119891119898 by ]

Let 1198911 119891119898 be BB-polynomials in 119896 + 1 variables with119873(119891119894) = 119875119894 and let Int(119875119894)⋂ Int(119875119895) = 0 for 119894 = 119895 Achart 119862119905119878(119891) of a BB-polynomial 119891 with 119873(119891) = 119875 is saidto be obtained by 119901119886119905119888ℎ119908119900119903119896119894119899119892 charts of BB-polynomials1198911 119891119898 and it is a 119901119886119905119888ℎ119908119900119903119896 of charts of BB-polynomials1198911 119891119898 if 119875 = ⋃

119898

119894=1119875119894 and the chart 119862119905119878(119891) of 119891 up to

isotopy is⋃119898

119894=1119862119905119878(119891119894)

The result about the Viro method for the constructionof the Bernstein-Bezier algebraic hypersurface piece on asimplex with a prescribed topology is shown in the followingproposition (see [18])

Proposition 3 (see [18]) Let 119875 1198751 119875119898 ] 1198911 119891119898 and119891119905 be as above (119891119905 is a119901119886119905119888ℎ119908119900119903119896 of BB-polynomials1198911 119891119898by ]) If BB-polynomials 1198911 119891119898 are nondegenerate thenthere exists 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart ofBB-polynomial 119891119905 is obtained by patchworking charts of BB-polynomials 1198911 119891119898

3 The Chart of the Piecewise Polynomial

In this section the chart of the piecewise polynomial isdefined and some properties of the chart of the Bernstein-Bezier polynomial and the chart of the piecewise polynomialare discussed

Theorem 4 Let 119891 be a BB-polynomial with Newton polyhe-dron 119875 and let Γ be a face of 119875 Then

(1) 119862119905119878(119891)⋂ Γ = 119862119905119878(119891Γ)

(2) if 119891Γ is nondegenerate with respect to Γ (which is thecase when for example119891 is nondegenerate with respectto 119875) then the 119862119905119878(119891) intersects Γ transversally

Proof Suppose that the BB-polynomial 119891 is defined in (4)and

119892 (119909) = sum

|120582|=119899

119899

1205821 sdot sdot sdot 120582119896+1

1198871205821199091205822

2sdot sdot sdot 119909

120582119896+1

119896+1 (9)

Define the mapping 120595 119878 rarr 119879(119896119899) = (1199092 119909119896+1) isin

(R+)119896|1199092 + sdot sdot sdot + 119909119896+1 le 119899 by

120595 (1205911 1205912 120591119896+1) = (1198991205912 119899120591119896+1)

(1205911 1205912 120591119896+1) isin 119878

(10)

According to [18 Lemma 32] then 120595(119875) is the Newtonpolyhedron 119873(119892) of the polynomial 119892 and 120595(Γ) is a face of119873(119892) if Γ is a face of 119875 Moreover by [10 Remark 11] the119888ℎ119886119903119905 119862119905+(119892) of 119892 (see [6ndash8 18]) and the truncation 119892120595(Γ) onthe face 120595(Γ) have the following properties

(a) 119862119905+(119892)⋂120595(Γ) = 119862119905+(119892120595(Γ))

(b) the 119888ℎ119886119903119905 119862119905+(119892) intersects 120595(Γ) transversally if 119892120595(Γ)

is nondegenerate with respect to 120595(Γ)

On the other hand by [18 Lemma 32 andTheorem 35]we can get the following equalities

120595minus1(119862119905+ (119892)⋂120595 (Γ)) = 119862119905119878 (119891)⋂Γ

120595minus1(119862119905+ (119892

120595(Γ))) = 119862119905119878 (119891

Γ)

(11)

This together with equality (10) and properties (a) and (b)shows that 119862119905119878(119891)⋂ Γ = 119862119905119878(119891

Γ) and the 119862119905119878(119891) intersects

Γ transversally if 119891Γ is nondegenerate with respect to Γ Thiscompletes the proof

The following result is a generalization of Farinrsquos theorem(see [3 15]) on high dimensional space

Proposition 5 (see [3 15]) Let

119901(1)(120591) = sum

|120582|=119899

119887(1)

120582119861120582119899 (120591) 119901

(2)(120591) = sum

|120582|=119899

119887(2)

120582119861120582119899 (120591)

(12)

be BB-polynomials of degree 119899 that are defined on two adjacent119896-dimensional simplices 1198781 = [V1 V2 V119896+1] and 1198782 =

[V10158401 V2 V119896+1] respectively Then 119901(1)(120591) and 119901(2)(120591) are 119862119903

smoothly connected on [V2 V119896+1] if and only if for all 120588 isin0 1 119903 and 1205822 120582119896+1 isin Z+ with sum

119896+1

119895=2120582119895 = 119899 minus 120588

119887(2)

120582120588= sum

|120598|=120588

119887(1)

120598+1205820119861120598120588 (120591

1015840) (13)

4 Abstract and Applied Analysis

where 1205820 = (0 1205822 120582119896+1) 120582120588 = (120588 1205822 120582119896+1) and 120598 =(1205981 1205982 120598119896+1) Here 1205911015840 denotes the barycentric coordinate ofV10158401with respect to the simplex 1198781

Theorem 6 Suppose that 119901(1)(120591) and 119901(2)(120591) defined as aboveare 119862119903

(119903 ge 0) smoothly connected on 1198781⋂1198782 = [V2 V119896+1]Then

(1) 119873(119901(1))⋂(1198781⋂1198782) = 119873(119901(2))⋂(1198781⋂1198782) and Γ =

119873(119901(1))⋂(1198781⋂1198782) is a face of 119873(119901(1)) and 119873(119901(2))

if 119873(119901(1))⋂(1198781⋂1198782) is nonempty where 119873(119901(1)) and119873(119901

(2)) are Newton polyhedra of BB-polynomials 119901(1)

and 119901(2) respectively(2) 119862119905119878

1

(119901(1))⋂ Γ = 119862119905119878

2

(119901(2))⋂ Γ for Γ above

Proof By assumption and taking 120588 = 0 in equality (13) wehave that 119887(1)

1205820= 119887

(2)

1205820for all 1205822 120582119896+1 isin Z+ with sum119896+1

119895=2120582119895 =

119899 Thus 120583(119901(1))⋂(1198781⋂1198782) = 120583(119901(2))⋂(1198781⋂1198782) where 120583

is defined in (6) This implies that 119873(119901(1))⋂(1198781⋂1198782) =

119873(119901(2))⋂(1198781⋂1198782) and so Γ = 119873(119901(1))⋂(1198781⋂1198782) is a face

of119873(119901(1)) and119873(119901(2)) if119873(119901(1))⋂(1198781⋂1198782) is nonemptyAccording to the argument above it is easy to see that Γ-

truncations (119901(1))Γ (119901(2))Γ of 119901(1) and 119901(2) satisfy the equality(119901

(1))Γ= (119901

(2))Γ Thus we can get that

1198621199051198781

((119901(1))

Γ

) = 1198621199051198782

((119901(2))

Γ

) (14)

by the definition of the 119888ℎ119886119903119905On the other hand it follows fromTheorem 4 that

1198621199051198781

(119901(1))⋂ Γ = 119862119905119878

1

((119901(1))

Γ

)

1198621199051198782

(119901(2))⋂ Γ = 119862119905119878

2

((119901(2))

Γ

)

(15)

Therefore

1198621199051198781

(119901(1))⋂Γ = 119862119905119878

2

(119901(2))⋂Γ (16)

by equality (14) This completes the proof

LetΔ be a pure hereditary simplicial complexwith 119896-cells(ie 119896-dimensional simplex) 1198781 119878119902 Let 119891 isin 119862

119903

119899(Δ) and

each polynomial 119891|119878120572

be expressed in the B-form

1198911003816100381610038161003816119878120572

(120591) = sum

|120582|=119899

119887[120572]

120582119861120582119899 (120591) 120572 = 1 2 119902 (17)

The piecewise polynomial 119891 defined as above is called 119862119903

piecewise BB-polynomial

Theorem 7 Let 119891 isin 119862119903

119899(Δ) be a piecewise BB-polynomial

defined in (17) Then for each adjacent pair 119878120572 119878120573 of 119896-cellsin Δ

(1) 119873(119891|119878120572

)⋂(119878120572⋂119878120573) = 119873(119891|119878120573

)⋂(119878120572⋂119878120573) andΓ120572120573 = 119873(119891|119878

120572

)⋂(119878120572⋂119878120573) is a face of 119873(119891|119878120572

) and119873(119891|119878

120573

) if 119873(119891|119878120572

)⋂(119878120572⋂119878120573) is nonempty where

119873(119891|119878120572

) and 119873(119891|119878120573

) are Newton polyhedra of BB-polynomials 119891|119878

120572

and 119891|119878120573

respectively

(2) 119862119905119878120572

(119891|119878120572

)⋂ Γ120572120573 = 119862119905119878120573

(119891|119878120573

)⋂ Γ120572120573 for Γ120572120573 above

Proof We can get the conclusion from Theorem 6 immedi-ately

According to Theorems 4ndash7 we can define the 119888ℎ119886119903119905 of apiecewise BB-polynomial

Definition 8 Let 119891 isin 119862119903

119899(Δ) be a piecewise BB-polynomial

defined in (17) and let 119873(119891|119878120572

) be the Newton polyhedronof the BB-polynomial 119891|119878

120572

The chart 119862119905(119891) of 119891 is theclosure of ⋃119902

120572=1120601120583(119891|

119878120572)(120591 isin Int(119878120572) 119891|119878

120572

(120591) = 0) where120583(119891|119878

120572

) = (1205821119899 120582119896+1119899)|120582 isin Z+(119896119899) 119887[120572]

120582= 0 and

120601120583(119891|119878120572) Int(119878120572) rarr Int(119873(119891|119878

120572

)) is the moment mapassociated with 120583(119891|119878

120572

) relative to the simplex 119878120572

Obviously we can get the following conclusion fromTheorems 4ndash7 and the definition above immediately

Theorem 9 Let 119891 isin 119862119903

119899(Δ) be a piecewise BB-polynomial

defined in (17) Then the 119888ℎ119886119903119905 119862119905(119891) of 119891 is⋃119902

120572=1119862119905119878120572

(119891|119878120572

)

4 The Construction of Real 119862119903 PiecewiseAlgebraic Hypersurfaces

In this section we propose a new method for the construc-tion of real piecewise algebraic hypersurfaces of a givendegree with certain smoothness and prescribed topologyThemethod is primarily based on the work of Lai et al in [18] andthe smooth blending theory

LetΔ be a pure hereditary simplicial complexwith 119896-cells(ie 119896-dimensional simplex) 1198781 119878119902

For each adjacent pair 119878120572 = [V(1205721) V(1205722) V(120572119896+1)]119878120573 = [V(1205731) V(1205732) V(120573119896+1)] of 119896-cells in Δ assume that119878120572 cap 119878120573 = [V(1205721) V(120572119894minus1) V(120572119894) V(120572119894+1) V(120572119896+1)] =

[V(1205731) V(120573119895minus1) V(120573119895) V(120573119895+1) V(120573119896+1)] where the hatmeans that the corresponding vertex is omitted Obvi-ously there exist two arrangements 1198971 1198972 119897119896+1 and 1198891 1198892 119889119896+1 of numbers 1 2 119896 + 1 such that V(120572119897

1) =

V(120572119894) V(1205731198891) = V(120573119895) and V(120572119897

119908) = V(120573119889

119908) for 119908 = 2 119896 + 1

Now we define two one-to-one transformations 120578120572120573120578120573120572 (R+)

119896+1rarr (R+)

119896+1 by 120578120572120573(120582) = (1205821198971

1205821198972

120582119897119896+1

)and 120578120573120572(120582) = (120582119889

1

1205821198892

120582119889119896+1

) respectively where 120582 =

(1205821 1205822 120582119896+1) isin (R+)119896+1

For any given 120572 isin 1 119902 let 119891(1205721) 119891

(120572119898120572) be BB-

polynomials over the simplex 119878120572 = [V(1205721) V(1205722) V(120572119896+1)]with119873(119891(120572120577)

) = 119875(120572120577) 120577 = 1 119898120572 and let

119891[120572]

119905(120591) = sum

|120582|=119899

119887[120572]

120582119861120582119899 (120591) 119905

][120572](120582119899)(18)

be a 119901119886119905119888ℎ119908119900119903119896119894119899119892 of BB-polynomials 119891(1205721) 119891

(120572119898120572) by

][120572]

Abstract and Applied Analysis 5

Define the piecewise BB-polynomial 119891119905 on the domainΩby

119891119905

1003816100381610038161003816119878120572

(120591) = 119891[120572]

119905(120591) 120572 = 1 2 119902 (19)

Theorem10 Suppose that119891(11) 119891

(11198981) 119891

(21) 119891

(119902119898119902)

are non-degenerate that 119891[120572]

119905is a 119901119886119905119888ℎ119908119900119903119896119894119899119892 of BB-

polynomials 119891(1205721) 119891

(120572119898120572) by ][120572] 120572 = 1 119902 and that

119878120572 119878120573 V(120572119894) V(120573119895) 120578120572120573 120578120573120572 119875(120572120577) 119891[120572]

119905 and 119891119905 are defined

as above and Int(119875(120572120577))⋂ Int(119875(120572120589)) = 0 for 120577 = 120589 If for eachadjacent pair 119878120572 119878120573 of 119896-cells in Δ and all 120588 isin 0 1 119903 and120582 = (1205821 1205822 120582119896+1) isin Z+(119896119899) with sum

119896+1

119894=2120582119894 = 119899 minus 120588

119887[120573]

(120578minus1

120573120572(120582120588))119905][120573](120578minus1

120573120572(120582120588)119899)

= sum

|120598|=120588

119887[120572]

120578minus1

120572120573(120598+1205820)119861120598120588 (120591

1015840

120573120572) 119905

][120572](120578minus1120572120573(120598+1205820)119899) 119905 isin R

lowast

+

(20)

where 120582120588 = (120588 1205822 120582119896+1) and 1205911015840120573120572 denotes the barycen-tric coordinate of V(120573119895) with respect to the simplex 119878120572 =

[V(1205721198971) V(120572119897

2) V(120572119897

119896+1)] then

(1) 119891119905 isin 119862119903

119899(Δ)

(2) if for any BB-polynomial 119891(120572119894) with 119873(119891(120572119894))⋂

(119878120572⋂119878120573) = 0 119894 isin 1 119898120572 there exists a BB-polynomial 119891(120573119895) in 119891

(1205731) 119891

(120573119898120573) such that

119873(119891(120573119895)

)⋂(119878120572⋂119878120573) = 119873(119891(120572119894))⋂(119878120572⋂119878120573) then

there is 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart119862119905(119891119905) of piecewise BB-polynomial 119891119905 up to isotopy is⋃

119902

120572=1⋃

119898120572

119894=1119862119905119878120572

(119891(120572119894))

Proof Let 120591 = (1205911 120591119896+1) and 120594 = (1205941 1205942 120594119896+1) bethe barycentric coordinates of point 119901 isin 119878120573 with respectto 119878120573 = [V(1205731) V(1205732) V(120573119896+1)] and 119878120573 = [V(120573119889

1) V(120573119889

2)

V(120573119889119896+1

)] respectively Then

120594 = (1205941 1205942 120594119896+1) = 120578120573120572 (120591) = (1205911198891

1205911198892

120591119889119896+1

) (21)

by the definition of the transformation 120578120573120572 and so

119891[120573]

119905(119901) = 119891

[120573]

119905 (120591)

= sum

|120582|=119899

119887[120573]

120582

119899

12058211205822 sdot sdot sdot 120582119896+1

times 1205911205821

11205911205822

2sdot sdot sdot 120591

120582119896+1

119896+1119905][120573](120582119899)

= sum

|120582|=119899

119887[120573]

120582

119899

1205821198891

1205821198891

sdot sdot sdot 120582119889119896+1

times (1205941)1205821198891(1205942)

1205821198892sdot sdot sdot (120594119896+1)

120582119889119896+1 119905

][120573](120582119899)

(22)

Set 120583 = (1205831 1205832 120583119896+1) = 120578120573120572(120582) Then 120582 = 120578minus1120573120572(120583) and

(1205831 1205832 120583119896+1) = (1205821198891

1205821198892

120582119889119896+1

) Therefore it followsfrom equality (22) that

119891[120573]

119905(119901) = sum

|120583|=119899

119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899)

times

119899

12058311205831 sdot sdot sdot 120583119896+1

(1205941)1205831

(1205942)1205832

sdot sdot sdot (120594119896+1)120583119896+1

(23)

This shows that the polynomial 119891[120573]

119905over 119878120573 = [V(120573119889

1) V(120573119889

2)

V(120573119889119896+1

)] can be represented as follows

119891[120573]

119905(120594) = sum

|120583|=119899

119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899)

119861120583119899 (120594) (24)

By a similar argument above we have that the polynomial119891[120572]

119905over 119878120572 = [V(120572119897

1) V(120572119897

119896+1)] can be represented as

follows

119891[120572]

119905(120594) = sum

|120583|=119899

119887[120572]

120578minus1

120572120573(120583)119905][120572](120578minus1

120572120573(120583)119899)

119861120583119899 (120594) (25)

where 120594 = (1205941 120594

2 120594

119896+1) is the barycentric coordinate of

point 119901 isin 119878120572 with respect to 119878120572 = [V(1205721198971) V(120572119897

2) V(120572119897

119896+1)]

Since V(120572119897119908) = V(120573119889

119908) for 119908 = 2 119896 + 1 119878120572⋂119878120573 =

[V(1205721198972) V(120572119897

119896+1)] thus according to Proposition 5 119891119905 isin

119862119903

119899(Δ) if and only if the BB-coefficients 119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899) of

119891[120573]

119905and the BB-coefficients 119887[120572]

120578minus1

120572120573(120583)119905][120572](120578minus1

120572120573(120583)119899) of 119891[120572]

119905satisfy

equality (20) This proves property (1)Below we will show that property (2) holdsFor each adjacent pair 119878120572 119878120573 of 119896-cells in Δ set Γ120572120573 =

119873(119891(120572119894))⋂(119878120572⋂119878120573)when119873(119891

(120572119894))⋂(119878120572⋂119878120573) = 0 then Γ120572120573

is a face of 119873(119891(120572119894)) and 119873(119891(120573119895)

) by the assumption that119873(119891

(120573119895))⋂(119878120572⋂119878120573) = 119873(119891

(120572119894))⋂(119878120572⋂119878120573) Since 119891[120572]

119905

(resp 119891[120573]

119905) is a patchworking of BB-polynomials 119891(1205721)

119891(120572119898120572) (resp 119891(1205731)

119891(120573119898120573)) by ][120572] (resp ][120573]) and

Int(119875(120572120577))⋂ Int(119875(120572120589)) = Int(119875(120573120577))⋂ Int(119875(120573120589)) = 0 for 120577 = 120589then for Γ120572120573-truncations (119891

(120572119894))Γ120572120573 (119891(120573119895)

)Γ120572120573 (119891[120572]

119905)Γ120572120573 and

(119891[120573]

119905)Γ120572120573 of 119891(120572119894) 119891(120573119895) 119891[120572]

119905 and 119891[120573]

119905 respectively we have

that

(119891(120572119894))

Γ120572120573

= (119891[120572]

119905)

Γ1205721205731003816100381610038161003816100381610038161003816119905=1

(119891(120573119895)

)

Γ120572120573

= (119891[120573]

119905)

Γ1205721205731003816100381610038161003816100381610038161003816119905=1

(26)

Since119891[120572]

119905and119891[120573]

119905are119862119903 smoothly connected on 119878120572⋂119878120573

119873(119891(120573119895)

)⋂(119878120572⋂119878120573) = 119873(119891(120572119894))⋂(119878120572⋂119878120573) and (20) implies

that (119891[120572]

119905)Γ120572120573= (119891

[120573]

119905)Γ120572120573 Thus (119891(120572119894)

)Γ120572120573= (119891

(120573119895))Γ120572120573 by (26)

6 Abstract and Applied Analysis

Using a similar argument in the proof of Theorem 6 wehave 119862119905119878

120572

(119891(120572119894))⋂ Γ120572120573 = 119862119905119878

120573

(119891(120573119895)

)⋂ Γ120572120573 by (119891(120572119894))Γ120572120573=

(119891(120573119895)

)Γ120572120573 This shows that

119898120572

⋃

119894=1

119862119905119878120572

(119891(120572119894))⋂(119878120572⋂119878120573)

=

119898120573

⋃

119895=1

119862119905119878120573

(119891(120573119895)

)⋂(119878120572⋂119878120573)

(27)

On the other hand since 119891[120572]

119905is a patchworking of

BB-polynomials 119891(1205721) 119891

(120572119898120572) by ][120572] we get from the

assumption and Proposition 3 that there is 119905[120572]0gt 0 such that

for any 119905 isin (0 119905[120572]0] the chart119862119905119878

120572

(119891[120572]

119905) of BB-polynomial119891[120572]

119905

(ie 119891119905|119878120572

) up to isotopy is⋃119898120572

119894=1119862119905119878120572

(119891(120572119894)) This conclusion

together with Theorems 7ndash9 and equality (27) implies thatthere is 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart119862119905(119891119905) of piecewise BB-polynomial 119891119905 up to isotopy is⋃

119902

120572=1⋃

119898120572

119894=1119862119905119878120572

(119891(120572119894)) where 1199050 = min119905[1]

0 119905

[119902]

0 This

completes the proof

Assume that 119875 is a 119896-dimensional simplex with domainpoints in 119878 as its vertices and that the BB-polynomial 119891 is areal 119896 + 1-nomial (this means that the nonzero coefficients in119891 correspond to the only vertices of 119875) Denote by119872(119875) theset of middle points of edges of 119875 For any point V isin 119872(119875) onan edge with endpoints (1198941119899 119894119896+1119899) (1198951119899 119895119896+1119899)we assign that

120575 (V) = sign (119887(1198941119894119896+1

) sdot 119887(1198951119895119896+1

)) (28)

where 119887(1198941119894119896+1

) 119887(1198951119895119896+1

) are the corresponding BB-coefficients in 119891 Put

119872minus (119875) = V isin 119872 (119875) | 120575 (V) = minus1 (29)

Theorem 11 Suppose that 119891(11) 119891

(11198981) 119891

(1199021)

119891(119902119898119902) 119875(11) 119875(1119898

1) 119875(1199021) 119875(119902119898

119902) 119891

[120572]

119905 and119891119905 sat-

isfy the conditions in Theorem 10 and that for each 120572 isin

1 2 119902 119891(1205721) 119891

(120572119898120572) are real (119896 + 1)-nomial and

119875(1205721) 119875(120572119898120572) with dimension 119896 form a triangulation of

119873(119891[120572]

119905) Then 119891119905 isin 119862119903

119899(Δ) and there exists a constant 1199050 gt 0

such that for any 119905 isin (0 1199050] the chart 119862119905(119891) of 119891 up to isotopyis ⋃119902

120572=1⋃

119898120572

119894=1conv(119872minus(119875(120572119894))) where 119872minus(119875(120572119894)) is defined as

(29)

Proof By assumption and [18 Theorem 44] we know thatthere is 119905[120572]

0gt 0 such that for any 119905 isin (0 119905

[120572]

0] the

chart 119862119905119878120572

(119891[120572]

119905) of BB-polynomial 119891[120572]

119905 up to isotopy is

⋃119898120572

119894=1conv(119872minus(119875(120572119894)))

The remainder of the proof can be completed by a similarapproach in the proof of Theorem 10

According toTheorems 10 and 11 if we want to constructa 119862119903 piecewise algebraic hypersurface 119891(120591) = 0 with aprescribed complex topology and a degree on a partition

Δ we can just construct some Bernstein-Bezier algebraichypersurfaces pieces 119891(1205721)

(120591) = 0 119891(120572119898120572)(120591) = 0 with

simple topology and a convex function ][120572] on each 119896-cell119878120572 where 119891

(1205721) 119891

(120572119898120572) and ][120572] satisfy the conditions

of Theorem 10 or Theorem 11 120572 = 1 2 119902 and then theBB-polynomial 119891[120572]

(120591) is obtained by a patchworking of BB-polynomials 119891(1205721)

119891(120572119898120572) by ][120572] Thus we can get 119891(120591)

by defining 119891|119878120572

(120591) = 119891[120572](120591)

Conflict of Interests

The authors do not have any possible conflict of interests inthis paper

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (Grant nos 11271328 11071031U0935004 U1135003 11290143 11101366 11226329 and61272307) the Zhejiang Provincial Natural Science Foun-dation (nos Y7080068 LQ13A010004) and the Foundationof Department of Education of Zhejiang Province (noY201223556)

References

[1] Y S Lai R H Wang and J M Wu ldquoReal zeros of the zero-dimensional parametric piecewise algebraic varietyrdquo Science inChina A vol 52 no 4 pp 817ndash832 2009

[2] Y S Lai R H Wang and J M Wu ldquoSolving parametricpiecewise polynomial systemsrdquo Journal of Computational andApplied Mathematics vol 236 no 5 pp 924ndash936 2011

[3] R-H Wang X Q Shi Z X Luo and Z X Su Multivariatespline and its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001

[4] O Ya Viro ldquoGluing of algebraic hypersurfaces smoothingof singularities and constructionof curvesrdquo in Proceedings ofthe Leningrad International Topological Conference pp 149ndash197Nauka Leningrad Russia 1983 (Russian)

[5] O Ya Viro ldquoGluing of plane real algebraic curves and construc-tions of curves of degrees 6 and 7rdquo inTopology (Leningrad 1982)vol 1060 of Lecture Notes inMathematics pp 187ndash200 SpringerBerlin Germany 1984

[6] O Ya Viro ldquoPatchworking real algebraic varietiesrdquo httpwwwmathsunysbedusimolegpwpdf

[7] E Shustin ldquoCritical points of real polynomials subdivisionsof Newton polyhedra and topology of real algebraic hypersur-facesrdquo Transactions of the American Mathematical Society vol173 no 2 pp 203ndash223 1996

[8] E Shustin ldquoGluing of singular and critical pointsrdquoTopology vol37 no 1 pp 195ndash217 1998

[9] E Shustin ldquoReal plane algebraic curves with prescribed singu-laritiesrdquo Topology vol 32 no 4 pp 845ndash856 1993

[10] F Bihan ldquoViro method for the construction of real completeintersectionsrdquo Advances in Mathematics vol 169 no 2 pp 177ndash186 2002

[11] J J Risler ldquoConstruction drsquohypersurfaces reelles [drsquoapres Viro]rdquoin Seminaire Bourbaki vol 199293 of Asterique no 216 Exp7633 pp 69ndash86 1993

Abstract and Applied Analysis 7

[12] I Itenberg and E Shustin ldquoSingular points and limit cycles ofplanar polynomial vector fieldsrdquo Duke Mathematical Journalvol 102 no 1 pp 1ndash37 2000

[13] I Itenberg and E Shustin ldquoViro theorem and topology of realand complex combinatorial hypersurfacesrdquo Israel Journal ofMathematics vol 133 no 1 pp 189ndash238 2003

[14] B Sturmfels ldquoVirorsquos theorem for complete intersectionsrdquoAnnalidella Scuola Normale Superiore di Pisa vol 21 no 3 pp 377ndash386 1994

[15] G Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Guide (Computer Science and ScientificComputing) Academic Press San Diego Calif USA 4thedition 1997

[16] C M Hoffmann Geometric and Solid Modeling Morgan Kauf-mann San Mateo Calif USA 1989

[17] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design AK Peters Boston Mass USA 1993

[18] Y S Lai W P Du and R H Wang ldquoThe Viro method forconstruction of Bernstein-Bezier algebraic hypersurface piecerdquoScience China Mathematics vol 55 no 6 pp 1269ndash1279 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

where Int(119875119878) is the complement in 119875119878 of the union of all itsproper faces

Definition 2 Let 119891 be a BB-polynomial with a Newtonpolyhedron 119875 Then the closure of 120601120583(119891)(120591 isin Int(119878) 119891(120591) =0) in 119875 is called chart 119862119905119878(119891) of 119891 where 120601120583(119891) Int(119878) rarrInt(119875) is themomentmap associated with 120583(119891) relative to thesimplex 119878

Let 119875 1198751 119875119898 sub 119878 be polyhedra with 119875 = ⋃119898

119894=1119875119894 and

Int(119875119894)⋂ Int(119875119895) = 0 for 119894 = 119895 Assume that ] 119875 rarr R isa continuous piecewise linear nonnegative convex functionsatisfying the following conditions

(1) all the restrictions ]|119875119894

are linear(2) if the restriction of ] to an open set is linear then this

set is contained in one of the 119875119894(3) ](119875⋂119871 (119896119899)) sub Z where 119871 (119896119899) is the set of the domain

points

Then this function ] with this property is said to convexify1198751 119875119898

Let1198911 119891119898 be BB-polynomials overR in 119896+1 variableswith119873(119891119894) = 119875119894 Let 119891

119875119894⋂119875119895

119894= 119891

119875119894⋂119875119895

119895for any 119894 119895 Then there

exists a unique BB-polynomial 119891 with 119873(119891) = 119875 = ⋃119898

119894=1119875119894

and 119891119875119894= 119891119894 for 119894 = 1 119898 If 119891(120591) = sum|120582|=119899

119887120582119861120582119899(120591) and ]is a function convexifying 1198751 P119898 we put

119891119905 (120591) = sum

|120582|=119899

119887120582119861120582119899 (120591) 119905](120582119899)

(8)

The BB-polynomials 119891119905 are said to be obtained by119901119886119905119888ℎ119908119900119903119896119894119899119892 BB-polynomials 1198911 119891119898 by ] or briefly 119891119905is a 119901119886119905119888ℎ119908119900119903119896 of BB-polynomials 1198911 119891119898 by ]

Let 1198911 119891119898 be BB-polynomials in 119896 + 1 variables with119873(119891119894) = 119875119894 and let Int(119875119894)⋂ Int(119875119895) = 0 for 119894 = 119895 Achart 119862119905119878(119891) of a BB-polynomial 119891 with 119873(119891) = 119875 is saidto be obtained by 119901119886119905119888ℎ119908119900119903119896119894119899119892 charts of BB-polynomials1198911 119891119898 and it is a 119901119886119905119888ℎ119908119900119903119896 of charts of BB-polynomials1198911 119891119898 if 119875 = ⋃

119898

119894=1119875119894 and the chart 119862119905119878(119891) of 119891 up to

isotopy is⋃119898

119894=1119862119905119878(119891119894)

The result about the Viro method for the constructionof the Bernstein-Bezier algebraic hypersurface piece on asimplex with a prescribed topology is shown in the followingproposition (see [18])

Proposition 3 (see [18]) Let 119875 1198751 119875119898 ] 1198911 119891119898 and119891119905 be as above (119891119905 is a119901119886119905119888ℎ119908119900119903119896 of BB-polynomials1198911 119891119898by ]) If BB-polynomials 1198911 119891119898 are nondegenerate thenthere exists 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart ofBB-polynomial 119891119905 is obtained by patchworking charts of BB-polynomials 1198911 119891119898

3 The Chart of the Piecewise Polynomial

In this section the chart of the piecewise polynomial isdefined and some properties of the chart of the Bernstein-Bezier polynomial and the chart of the piecewise polynomialare discussed

Theorem 4 Let 119891 be a BB-polynomial with Newton polyhe-dron 119875 and let Γ be a face of 119875 Then

(1) 119862119905119878(119891)⋂ Γ = 119862119905119878(119891Γ)

(2) if 119891Γ is nondegenerate with respect to Γ (which is thecase when for example119891 is nondegenerate with respectto 119875) then the 119862119905119878(119891) intersects Γ transversally

Proof Suppose that the BB-polynomial 119891 is defined in (4)and

119892 (119909) = sum

|120582|=119899

119899

1205821 sdot sdot sdot 120582119896+1

1198871205821199091205822

2sdot sdot sdot 119909

120582119896+1

119896+1 (9)

Define the mapping 120595 119878 rarr 119879(119896119899) = (1199092 119909119896+1) isin

(R+)119896|1199092 + sdot sdot sdot + 119909119896+1 le 119899 by

120595 (1205911 1205912 120591119896+1) = (1198991205912 119899120591119896+1)

(1205911 1205912 120591119896+1) isin 119878

(10)

According to [18 Lemma 32] then 120595(119875) is the Newtonpolyhedron 119873(119892) of the polynomial 119892 and 120595(Γ) is a face of119873(119892) if Γ is a face of 119875 Moreover by [10 Remark 11] the119888ℎ119886119903119905 119862119905+(119892) of 119892 (see [6ndash8 18]) and the truncation 119892120595(Γ) onthe face 120595(Γ) have the following properties

(a) 119862119905+(119892)⋂120595(Γ) = 119862119905+(119892120595(Γ))

(b) the 119888ℎ119886119903119905 119862119905+(119892) intersects 120595(Γ) transversally if 119892120595(Γ)

is nondegenerate with respect to 120595(Γ)

On the other hand by [18 Lemma 32 andTheorem 35]we can get the following equalities

120595minus1(119862119905+ (119892)⋂120595 (Γ)) = 119862119905119878 (119891)⋂Γ

120595minus1(119862119905+ (119892

120595(Γ))) = 119862119905119878 (119891

Γ)

(11)

This together with equality (10) and properties (a) and (b)shows that 119862119905119878(119891)⋂ Γ = 119862119905119878(119891

Γ) and the 119862119905119878(119891) intersects

Γ transversally if 119891Γ is nondegenerate with respect to Γ Thiscompletes the proof

The following result is a generalization of Farinrsquos theorem(see [3 15]) on high dimensional space

Proposition 5 (see [3 15]) Let

119901(1)(120591) = sum

|120582|=119899

119887(1)

120582119861120582119899 (120591) 119901

(2)(120591) = sum

|120582|=119899

119887(2)

120582119861120582119899 (120591)

(12)

be BB-polynomials of degree 119899 that are defined on two adjacent119896-dimensional simplices 1198781 = [V1 V2 V119896+1] and 1198782 =

[V10158401 V2 V119896+1] respectively Then 119901(1)(120591) and 119901(2)(120591) are 119862119903

smoothly connected on [V2 V119896+1] if and only if for all 120588 isin0 1 119903 and 1205822 120582119896+1 isin Z+ with sum

119896+1

119895=2120582119895 = 119899 minus 120588

119887(2)

120582120588= sum

|120598|=120588

119887(1)

120598+1205820119861120598120588 (120591

1015840) (13)

4 Abstract and Applied Analysis

where 1205820 = (0 1205822 120582119896+1) 120582120588 = (120588 1205822 120582119896+1) and 120598 =(1205981 1205982 120598119896+1) Here 1205911015840 denotes the barycentric coordinate ofV10158401with respect to the simplex 1198781

Theorem 6 Suppose that 119901(1)(120591) and 119901(2)(120591) defined as aboveare 119862119903

(119903 ge 0) smoothly connected on 1198781⋂1198782 = [V2 V119896+1]Then

(1) 119873(119901(1))⋂(1198781⋂1198782) = 119873(119901(2))⋂(1198781⋂1198782) and Γ =

119873(119901(1))⋂(1198781⋂1198782) is a face of 119873(119901(1)) and 119873(119901(2))

if 119873(119901(1))⋂(1198781⋂1198782) is nonempty where 119873(119901(1)) and119873(119901

(2)) are Newton polyhedra of BB-polynomials 119901(1)

and 119901(2) respectively(2) 119862119905119878

1

(119901(1))⋂ Γ = 119862119905119878

2

(119901(2))⋂ Γ for Γ above

Proof By assumption and taking 120588 = 0 in equality (13) wehave that 119887(1)

1205820= 119887

(2)

1205820for all 1205822 120582119896+1 isin Z+ with sum119896+1

119895=2120582119895 =

119899 Thus 120583(119901(1))⋂(1198781⋂1198782) = 120583(119901(2))⋂(1198781⋂1198782) where 120583

is defined in (6) This implies that 119873(119901(1))⋂(1198781⋂1198782) =

119873(119901(2))⋂(1198781⋂1198782) and so Γ = 119873(119901(1))⋂(1198781⋂1198782) is a face

of119873(119901(1)) and119873(119901(2)) if119873(119901(1))⋂(1198781⋂1198782) is nonemptyAccording to the argument above it is easy to see that Γ-

truncations (119901(1))Γ (119901(2))Γ of 119901(1) and 119901(2) satisfy the equality(119901

(1))Γ= (119901

(2))Γ Thus we can get that

1198621199051198781

((119901(1))

Γ

) = 1198621199051198782

((119901(2))

Γ

) (14)

by the definition of the 119888ℎ119886119903119905On the other hand it follows fromTheorem 4 that

1198621199051198781

(119901(1))⋂ Γ = 119862119905119878

1

((119901(1))

Γ

)

1198621199051198782

(119901(2))⋂ Γ = 119862119905119878

2

((119901(2))

Γ

)

(15)

Therefore

1198621199051198781

(119901(1))⋂Γ = 119862119905119878

2

(119901(2))⋂Γ (16)

by equality (14) This completes the proof

LetΔ be a pure hereditary simplicial complexwith 119896-cells(ie 119896-dimensional simplex) 1198781 119878119902 Let 119891 isin 119862

119903

119899(Δ) and

each polynomial 119891|119878120572

be expressed in the B-form

1198911003816100381610038161003816119878120572

(120591) = sum

|120582|=119899

119887[120572]

120582119861120582119899 (120591) 120572 = 1 2 119902 (17)

The piecewise polynomial 119891 defined as above is called 119862119903

piecewise BB-polynomial

Theorem 7 Let 119891 isin 119862119903

119899(Δ) be a piecewise BB-polynomial

defined in (17) Then for each adjacent pair 119878120572 119878120573 of 119896-cellsin Δ

(1) 119873(119891|119878120572

)⋂(119878120572⋂119878120573) = 119873(119891|119878120573

)⋂(119878120572⋂119878120573) andΓ120572120573 = 119873(119891|119878

120572

)⋂(119878120572⋂119878120573) is a face of 119873(119891|119878120572

) and119873(119891|119878

120573

) if 119873(119891|119878120572

)⋂(119878120572⋂119878120573) is nonempty where

119873(119891|119878120572

) and 119873(119891|119878120573

) are Newton polyhedra of BB-polynomials 119891|119878

120572

and 119891|119878120573

respectively

(2) 119862119905119878120572

(119891|119878120572

)⋂ Γ120572120573 = 119862119905119878120573

(119891|119878120573

)⋂ Γ120572120573 for Γ120572120573 above

Proof We can get the conclusion from Theorem 6 immedi-ately

According to Theorems 4ndash7 we can define the 119888ℎ119886119903119905 of apiecewise BB-polynomial

Definition 8 Let 119891 isin 119862119903

119899(Δ) be a piecewise BB-polynomial

defined in (17) and let 119873(119891|119878120572

) be the Newton polyhedronof the BB-polynomial 119891|119878

120572

The chart 119862119905(119891) of 119891 is theclosure of ⋃119902

120572=1120601120583(119891|

119878120572)(120591 isin Int(119878120572) 119891|119878

120572

(120591) = 0) where120583(119891|119878

120572

) = (1205821119899 120582119896+1119899)|120582 isin Z+(119896119899) 119887[120572]

120582= 0 and

120601120583(119891|119878120572) Int(119878120572) rarr Int(119873(119891|119878

120572

)) is the moment mapassociated with 120583(119891|119878

120572

) relative to the simplex 119878120572

Obviously we can get the following conclusion fromTheorems 4ndash7 and the definition above immediately

Theorem 9 Let 119891 isin 119862119903

119899(Δ) be a piecewise BB-polynomial

defined in (17) Then the 119888ℎ119886119903119905 119862119905(119891) of 119891 is⋃119902

120572=1119862119905119878120572

(119891|119878120572

)

4 The Construction of Real 119862119903 PiecewiseAlgebraic Hypersurfaces

In this section we propose a new method for the construc-tion of real piecewise algebraic hypersurfaces of a givendegree with certain smoothness and prescribed topologyThemethod is primarily based on the work of Lai et al in [18] andthe smooth blending theory

LetΔ be a pure hereditary simplicial complexwith 119896-cells(ie 119896-dimensional simplex) 1198781 119878119902

For each adjacent pair 119878120572 = [V(1205721) V(1205722) V(120572119896+1)]119878120573 = [V(1205731) V(1205732) V(120573119896+1)] of 119896-cells in Δ assume that119878120572 cap 119878120573 = [V(1205721) V(120572119894minus1) V(120572119894) V(120572119894+1) V(120572119896+1)] =

[V(1205731) V(120573119895minus1) V(120573119895) V(120573119895+1) V(120573119896+1)] where the hatmeans that the corresponding vertex is omitted Obvi-ously there exist two arrangements 1198971 1198972 119897119896+1 and 1198891 1198892 119889119896+1 of numbers 1 2 119896 + 1 such that V(120572119897

1) =

V(120572119894) V(1205731198891) = V(120573119895) and V(120572119897

119908) = V(120573119889

119908) for 119908 = 2 119896 + 1

Now we define two one-to-one transformations 120578120572120573120578120573120572 (R+)

119896+1rarr (R+)

119896+1 by 120578120572120573(120582) = (1205821198971

1205821198972

120582119897119896+1

)and 120578120573120572(120582) = (120582119889

1

1205821198892

120582119889119896+1

) respectively where 120582 =

(1205821 1205822 120582119896+1) isin (R+)119896+1

For any given 120572 isin 1 119902 let 119891(1205721) 119891

(120572119898120572) be BB-

polynomials over the simplex 119878120572 = [V(1205721) V(1205722) V(120572119896+1)]with119873(119891(120572120577)

) = 119875(120572120577) 120577 = 1 119898120572 and let

119891[120572]

119905(120591) = sum

|120582|=119899

119887[120572]

120582119861120582119899 (120591) 119905

][120572](120582119899)(18)

be a 119901119886119905119888ℎ119908119900119903119896119894119899119892 of BB-polynomials 119891(1205721) 119891

(120572119898120572) by

][120572]

Abstract and Applied Analysis 5

Define the piecewise BB-polynomial 119891119905 on the domainΩby

119891119905

1003816100381610038161003816119878120572

(120591) = 119891[120572]

119905(120591) 120572 = 1 2 119902 (19)

Theorem10 Suppose that119891(11) 119891

(11198981) 119891

(21) 119891

(119902119898119902)

are non-degenerate that 119891[120572]

119905is a 119901119886119905119888ℎ119908119900119903119896119894119899119892 of BB-

polynomials 119891(1205721) 119891

(120572119898120572) by ][120572] 120572 = 1 119902 and that

119878120572 119878120573 V(120572119894) V(120573119895) 120578120572120573 120578120573120572 119875(120572120577) 119891[120572]

119905 and 119891119905 are defined

as above and Int(119875(120572120577))⋂ Int(119875(120572120589)) = 0 for 120577 = 120589 If for eachadjacent pair 119878120572 119878120573 of 119896-cells in Δ and all 120588 isin 0 1 119903 and120582 = (1205821 1205822 120582119896+1) isin Z+(119896119899) with sum

119896+1

119894=2120582119894 = 119899 minus 120588

119887[120573]

(120578minus1

120573120572(120582120588))119905][120573](120578minus1

120573120572(120582120588)119899)

= sum

|120598|=120588

119887[120572]

120578minus1

120572120573(120598+1205820)119861120598120588 (120591

1015840

120573120572) 119905

][120572](120578minus1120572120573(120598+1205820)119899) 119905 isin R

lowast

+

(20)

where 120582120588 = (120588 1205822 120582119896+1) and 1205911015840120573120572 denotes the barycen-tric coordinate of V(120573119895) with respect to the simplex 119878120572 =

[V(1205721198971) V(120572119897

2) V(120572119897

119896+1)] then

(1) 119891119905 isin 119862119903

119899(Δ)

(2) if for any BB-polynomial 119891(120572119894) with 119873(119891(120572119894))⋂

(119878120572⋂119878120573) = 0 119894 isin 1 119898120572 there exists a BB-polynomial 119891(120573119895) in 119891

(1205731) 119891

(120573119898120573) such that

119873(119891(120573119895)

)⋂(119878120572⋂119878120573) = 119873(119891(120572119894))⋂(119878120572⋂119878120573) then

there is 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart119862119905(119891119905) of piecewise BB-polynomial 119891119905 up to isotopy is⋃

119902

120572=1⋃

119898120572

119894=1119862119905119878120572

(119891(120572119894))

Proof Let 120591 = (1205911 120591119896+1) and 120594 = (1205941 1205942 120594119896+1) bethe barycentric coordinates of point 119901 isin 119878120573 with respectto 119878120573 = [V(1205731) V(1205732) V(120573119896+1)] and 119878120573 = [V(120573119889

1) V(120573119889

2)

V(120573119889119896+1

)] respectively Then

120594 = (1205941 1205942 120594119896+1) = 120578120573120572 (120591) = (1205911198891

1205911198892

120591119889119896+1

) (21)

by the definition of the transformation 120578120573120572 and so

119891[120573]

119905(119901) = 119891

[120573]

119905 (120591)

= sum

|120582|=119899

119887[120573]

120582

119899

12058211205822 sdot sdot sdot 120582119896+1

times 1205911205821

11205911205822

2sdot sdot sdot 120591

120582119896+1

119896+1119905][120573](120582119899)

= sum

|120582|=119899

119887[120573]

120582

119899

1205821198891

1205821198891

sdot sdot sdot 120582119889119896+1

times (1205941)1205821198891(1205942)

1205821198892sdot sdot sdot (120594119896+1)

120582119889119896+1 119905

][120573](120582119899)

(22)

Set 120583 = (1205831 1205832 120583119896+1) = 120578120573120572(120582) Then 120582 = 120578minus1120573120572(120583) and

(1205831 1205832 120583119896+1) = (1205821198891

1205821198892

120582119889119896+1

) Therefore it followsfrom equality (22) that

119891[120573]

119905(119901) = sum

|120583|=119899

119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899)

times

119899

12058311205831 sdot sdot sdot 120583119896+1

(1205941)1205831

(1205942)1205832

sdot sdot sdot (120594119896+1)120583119896+1

(23)

This shows that the polynomial 119891[120573]

119905over 119878120573 = [V(120573119889

1) V(120573119889

2)

V(120573119889119896+1

)] can be represented as follows

119891[120573]

119905(120594) = sum

|120583|=119899

119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899)

119861120583119899 (120594) (24)

By a similar argument above we have that the polynomial119891[120572]

119905over 119878120572 = [V(120572119897

1) V(120572119897

119896+1)] can be represented as

follows

119891[120572]

119905(120594) = sum

|120583|=119899

119887[120572]

120578minus1

120572120573(120583)119905][120572](120578minus1

120572120573(120583)119899)

119861120583119899 (120594) (25)

where 120594 = (1205941 120594

2 120594

119896+1) is the barycentric coordinate of

point 119901 isin 119878120572 with respect to 119878120572 = [V(1205721198971) V(120572119897

2) V(120572119897

119896+1)]

Since V(120572119897119908) = V(120573119889

119908) for 119908 = 2 119896 + 1 119878120572⋂119878120573 =

[V(1205721198972) V(120572119897

119896+1)] thus according to Proposition 5 119891119905 isin

119862119903

119899(Δ) if and only if the BB-coefficients 119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899) of

119891[120573]

119905and the BB-coefficients 119887[120572]

120578minus1

120572120573(120583)119905][120572](120578minus1

120572120573(120583)119899) of 119891[120572]

119905satisfy

equality (20) This proves property (1)Below we will show that property (2) holdsFor each adjacent pair 119878120572 119878120573 of 119896-cells in Δ set Γ120572120573 =

119873(119891(120572119894))⋂(119878120572⋂119878120573)when119873(119891

(120572119894))⋂(119878120572⋂119878120573) = 0 then Γ120572120573

is a face of 119873(119891(120572119894)) and 119873(119891(120573119895)

) by the assumption that119873(119891

(120573119895))⋂(119878120572⋂119878120573) = 119873(119891

(120572119894))⋂(119878120572⋂119878120573) Since 119891[120572]

119905

(resp 119891[120573]

119905) is a patchworking of BB-polynomials 119891(1205721)

119891(120572119898120572) (resp 119891(1205731)

119891(120573119898120573)) by ][120572] (resp ][120573]) and

Int(119875(120572120577))⋂ Int(119875(120572120589)) = Int(119875(120573120577))⋂ Int(119875(120573120589)) = 0 for 120577 = 120589then for Γ120572120573-truncations (119891

(120572119894))Γ120572120573 (119891(120573119895)

)Γ120572120573 (119891[120572]

119905)Γ120572120573 and

(119891[120573]

119905)Γ120572120573 of 119891(120572119894) 119891(120573119895) 119891[120572]

119905 and 119891[120573]

119905 respectively we have

that

(119891(120572119894))

Γ120572120573

= (119891[120572]

119905)

Γ1205721205731003816100381610038161003816100381610038161003816119905=1

(119891(120573119895)

)

Γ120572120573

= (119891[120573]

119905)

Γ1205721205731003816100381610038161003816100381610038161003816119905=1

(26)

Since119891[120572]

119905and119891[120573]

119905are119862119903 smoothly connected on 119878120572⋂119878120573

119873(119891(120573119895)

)⋂(119878120572⋂119878120573) = 119873(119891(120572119894))⋂(119878120572⋂119878120573) and (20) implies

that (119891[120572]

119905)Γ120572120573= (119891

[120573]

119905)Γ120572120573 Thus (119891(120572119894)

)Γ120572120573= (119891

(120573119895))Γ120572120573 by (26)

6 Abstract and Applied Analysis

Using a similar argument in the proof of Theorem 6 wehave 119862119905119878

120572

(119891(120572119894))⋂ Γ120572120573 = 119862119905119878

120573

(119891(120573119895)

)⋂ Γ120572120573 by (119891(120572119894))Γ120572120573=

(119891(120573119895)

)Γ120572120573 This shows that

119898120572

⋃

119894=1

119862119905119878120572

(119891(120572119894))⋂(119878120572⋂119878120573)

=

119898120573

⋃

119895=1

119862119905119878120573

(119891(120573119895)

)⋂(119878120572⋂119878120573)

(27)

On the other hand since 119891[120572]

119905is a patchworking of

BB-polynomials 119891(1205721) 119891

(120572119898120572) by ][120572] we get from the

assumption and Proposition 3 that there is 119905[120572]0gt 0 such that

for any 119905 isin (0 119905[120572]0] the chart119862119905119878

120572

(119891[120572]

119905) of BB-polynomial119891[120572]

119905

(ie 119891119905|119878120572

) up to isotopy is⋃119898120572

119894=1119862119905119878120572

(119891(120572119894)) This conclusion

together with Theorems 7ndash9 and equality (27) implies thatthere is 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart119862119905(119891119905) of piecewise BB-polynomial 119891119905 up to isotopy is⋃

119902

120572=1⋃

119898120572

119894=1119862119905119878120572

(119891(120572119894)) where 1199050 = min119905[1]

0 119905

[119902]

0 This

completes the proof

Assume that 119875 is a 119896-dimensional simplex with domainpoints in 119878 as its vertices and that the BB-polynomial 119891 is areal 119896 + 1-nomial (this means that the nonzero coefficients in119891 correspond to the only vertices of 119875) Denote by119872(119875) theset of middle points of edges of 119875 For any point V isin 119872(119875) onan edge with endpoints (1198941119899 119894119896+1119899) (1198951119899 119895119896+1119899)we assign that

120575 (V) = sign (119887(1198941119894119896+1

) sdot 119887(1198951119895119896+1

)) (28)

where 119887(1198941119894119896+1

) 119887(1198951119895119896+1

) are the corresponding BB-coefficients in 119891 Put

119872minus (119875) = V isin 119872 (119875) | 120575 (V) = minus1 (29)

Theorem 11 Suppose that 119891(11) 119891

(11198981) 119891

(1199021)

119891(119902119898119902) 119875(11) 119875(1119898

1) 119875(1199021) 119875(119902119898

119902) 119891

[120572]

119905 and119891119905 sat-

isfy the conditions in Theorem 10 and that for each 120572 isin

1 2 119902 119891(1205721) 119891

(120572119898120572) are real (119896 + 1)-nomial and

119875(1205721) 119875(120572119898120572) with dimension 119896 form a triangulation of

119873(119891[120572]

119905) Then 119891119905 isin 119862119903

119899(Δ) and there exists a constant 1199050 gt 0

such that for any 119905 isin (0 1199050] the chart 119862119905(119891) of 119891 up to isotopyis ⋃119902

120572=1⋃

119898120572

119894=1conv(119872minus(119875(120572119894))) where 119872minus(119875(120572119894)) is defined as

(29)

Proof By assumption and [18 Theorem 44] we know thatthere is 119905[120572]

0gt 0 such that for any 119905 isin (0 119905

[120572]

0] the

chart 119862119905119878120572

(119891[120572]

119905) of BB-polynomial 119891[120572]

119905 up to isotopy is

⋃119898120572

119894=1conv(119872minus(119875(120572119894)))

The remainder of the proof can be completed by a similarapproach in the proof of Theorem 10

According toTheorems 10 and 11 if we want to constructa 119862119903 piecewise algebraic hypersurface 119891(120591) = 0 with aprescribed complex topology and a degree on a partition

Δ we can just construct some Bernstein-Bezier algebraichypersurfaces pieces 119891(1205721)

(120591) = 0 119891(120572119898120572)(120591) = 0 with

simple topology and a convex function ][120572] on each 119896-cell119878120572 where 119891

(1205721) 119891

(120572119898120572) and ][120572] satisfy the conditions

of Theorem 10 or Theorem 11 120572 = 1 2 119902 and then theBB-polynomial 119891[120572]

(120591) is obtained by a patchworking of BB-polynomials 119891(1205721)

119891(120572119898120572) by ][120572] Thus we can get 119891(120591)

by defining 119891|119878120572

(120591) = 119891[120572](120591)

Conflict of Interests

The authors do not have any possible conflict of interests inthis paper

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (Grant nos 11271328 11071031U0935004 U1135003 11290143 11101366 11226329 and61272307) the Zhejiang Provincial Natural Science Foun-dation (nos Y7080068 LQ13A010004) and the Foundationof Department of Education of Zhejiang Province (noY201223556)

References

[1] Y S Lai R H Wang and J M Wu ldquoReal zeros of the zero-dimensional parametric piecewise algebraic varietyrdquo Science inChina A vol 52 no 4 pp 817ndash832 2009

[2] Y S Lai R H Wang and J M Wu ldquoSolving parametricpiecewise polynomial systemsrdquo Journal of Computational andApplied Mathematics vol 236 no 5 pp 924ndash936 2011

[3] R-H Wang X Q Shi Z X Luo and Z X Su Multivariatespline and its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001

[4] O Ya Viro ldquoGluing of algebraic hypersurfaces smoothingof singularities and constructionof curvesrdquo in Proceedings ofthe Leningrad International Topological Conference pp 149ndash197Nauka Leningrad Russia 1983 (Russian)

[5] O Ya Viro ldquoGluing of plane real algebraic curves and construc-tions of curves of degrees 6 and 7rdquo inTopology (Leningrad 1982)vol 1060 of Lecture Notes inMathematics pp 187ndash200 SpringerBerlin Germany 1984

[6] O Ya Viro ldquoPatchworking real algebraic varietiesrdquo httpwwwmathsunysbedusimolegpwpdf

[7] E Shustin ldquoCritical points of real polynomials subdivisionsof Newton polyhedra and topology of real algebraic hypersur-facesrdquo Transactions of the American Mathematical Society vol173 no 2 pp 203ndash223 1996

[8] E Shustin ldquoGluing of singular and critical pointsrdquoTopology vol37 no 1 pp 195ndash217 1998

[9] E Shustin ldquoReal plane algebraic curves with prescribed singu-laritiesrdquo Topology vol 32 no 4 pp 845ndash856 1993

[10] F Bihan ldquoViro method for the construction of real completeintersectionsrdquo Advances in Mathematics vol 169 no 2 pp 177ndash186 2002

[11] J J Risler ldquoConstruction drsquohypersurfaces reelles [drsquoapres Viro]rdquoin Seminaire Bourbaki vol 199293 of Asterique no 216 Exp7633 pp 69ndash86 1993

Abstract and Applied Analysis 7

[12] I Itenberg and E Shustin ldquoSingular points and limit cycles ofplanar polynomial vector fieldsrdquo Duke Mathematical Journalvol 102 no 1 pp 1ndash37 2000

[13] I Itenberg and E Shustin ldquoViro theorem and topology of realand complex combinatorial hypersurfacesrdquo Israel Journal ofMathematics vol 133 no 1 pp 189ndash238 2003

[14] B Sturmfels ldquoVirorsquos theorem for complete intersectionsrdquoAnnalidella Scuola Normale Superiore di Pisa vol 21 no 3 pp 377ndash386 1994

[15] G Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Guide (Computer Science and ScientificComputing) Academic Press San Diego Calif USA 4thedition 1997

[16] C M Hoffmann Geometric and Solid Modeling Morgan Kauf-mann San Mateo Calif USA 1989

[17] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design AK Peters Boston Mass USA 1993

[18] Y S Lai W P Du and R H Wang ldquoThe Viro method forconstruction of Bernstein-Bezier algebraic hypersurface piecerdquoScience China Mathematics vol 55 no 6 pp 1269ndash1279 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

where 1205820 = (0 1205822 120582119896+1) 120582120588 = (120588 1205822 120582119896+1) and 120598 =(1205981 1205982 120598119896+1) Here 1205911015840 denotes the barycentric coordinate ofV10158401with respect to the simplex 1198781

Theorem 6 Suppose that 119901(1)(120591) and 119901(2)(120591) defined as aboveare 119862119903

(119903 ge 0) smoothly connected on 1198781⋂1198782 = [V2 V119896+1]Then

(1) 119873(119901(1))⋂(1198781⋂1198782) = 119873(119901(2))⋂(1198781⋂1198782) and Γ =

119873(119901(1))⋂(1198781⋂1198782) is a face of 119873(119901(1)) and 119873(119901(2))

if 119873(119901(1))⋂(1198781⋂1198782) is nonempty where 119873(119901(1)) and119873(119901

(2)) are Newton polyhedra of BB-polynomials 119901(1)

and 119901(2) respectively(2) 119862119905119878

1

(119901(1))⋂ Γ = 119862119905119878

2

(119901(2))⋂ Γ for Γ above

Proof By assumption and taking 120588 = 0 in equality (13) wehave that 119887(1)

1205820= 119887

(2)

1205820for all 1205822 120582119896+1 isin Z+ with sum119896+1

119895=2120582119895 =

119899 Thus 120583(119901(1))⋂(1198781⋂1198782) = 120583(119901(2))⋂(1198781⋂1198782) where 120583

is defined in (6) This implies that 119873(119901(1))⋂(1198781⋂1198782) =

119873(119901(2))⋂(1198781⋂1198782) and so Γ = 119873(119901(1))⋂(1198781⋂1198782) is a face

of119873(119901(1)) and119873(119901(2)) if119873(119901(1))⋂(1198781⋂1198782) is nonemptyAccording to the argument above it is easy to see that Γ-

truncations (119901(1))Γ (119901(2))Γ of 119901(1) and 119901(2) satisfy the equality(119901

(1))Γ= (119901

(2))Γ Thus we can get that

1198621199051198781

((119901(1))

Γ

) = 1198621199051198782

((119901(2))

Γ

) (14)

by the definition of the 119888ℎ119886119903119905On the other hand it follows fromTheorem 4 that

1198621199051198781

(119901(1))⋂ Γ = 119862119905119878

1

((119901(1))

Γ

)

1198621199051198782

(119901(2))⋂ Γ = 119862119905119878

2

((119901(2))

Γ

)

(15)

Therefore

1198621199051198781

(119901(1))⋂Γ = 119862119905119878

2

(119901(2))⋂Γ (16)

by equality (14) This completes the proof

LetΔ be a pure hereditary simplicial complexwith 119896-cells(ie 119896-dimensional simplex) 1198781 119878119902 Let 119891 isin 119862

119903

119899(Δ) and

each polynomial 119891|119878120572

be expressed in the B-form

1198911003816100381610038161003816119878120572

(120591) = sum

|120582|=119899

119887[120572]

120582119861120582119899 (120591) 120572 = 1 2 119902 (17)

The piecewise polynomial 119891 defined as above is called 119862119903

piecewise BB-polynomial

Theorem 7 Let 119891 isin 119862119903

119899(Δ) be a piecewise BB-polynomial

defined in (17) Then for each adjacent pair 119878120572 119878120573 of 119896-cellsin Δ

(1) 119873(119891|119878120572

)⋂(119878120572⋂119878120573) = 119873(119891|119878120573

)⋂(119878120572⋂119878120573) andΓ120572120573 = 119873(119891|119878

120572

)⋂(119878120572⋂119878120573) is a face of 119873(119891|119878120572

) and119873(119891|119878

120573

) if 119873(119891|119878120572

)⋂(119878120572⋂119878120573) is nonempty where

119873(119891|119878120572

) and 119873(119891|119878120573

) are Newton polyhedra of BB-polynomials 119891|119878

120572

and 119891|119878120573

respectively

(2) 119862119905119878120572

(119891|119878120572

)⋂ Γ120572120573 = 119862119905119878120573

(119891|119878120573

)⋂ Γ120572120573 for Γ120572120573 above

Proof We can get the conclusion from Theorem 6 immedi-ately

According to Theorems 4ndash7 we can define the 119888ℎ119886119903119905 of apiecewise BB-polynomial

Definition 8 Let 119891 isin 119862119903

119899(Δ) be a piecewise BB-polynomial

defined in (17) and let 119873(119891|119878120572

) be the Newton polyhedronof the BB-polynomial 119891|119878

120572

The chart 119862119905(119891) of 119891 is theclosure of ⋃119902

120572=1120601120583(119891|

119878120572)(120591 isin Int(119878120572) 119891|119878

120572

(120591) = 0) where120583(119891|119878

120572

) = (1205821119899 120582119896+1119899)|120582 isin Z+(119896119899) 119887[120572]

120582= 0 and

120601120583(119891|119878120572) Int(119878120572) rarr Int(119873(119891|119878

120572

)) is the moment mapassociated with 120583(119891|119878

120572

) relative to the simplex 119878120572

Obviously we can get the following conclusion fromTheorems 4ndash7 and the definition above immediately

Theorem 9 Let 119891 isin 119862119903

119899(Δ) be a piecewise BB-polynomial

defined in (17) Then the 119888ℎ119886119903119905 119862119905(119891) of 119891 is⋃119902

120572=1119862119905119878120572

(119891|119878120572

)

4 The Construction of Real 119862119903 PiecewiseAlgebraic Hypersurfaces

In this section we propose a new method for the construc-tion of real piecewise algebraic hypersurfaces of a givendegree with certain smoothness and prescribed topologyThemethod is primarily based on the work of Lai et al in [18] andthe smooth blending theory

LetΔ be a pure hereditary simplicial complexwith 119896-cells(ie 119896-dimensional simplex) 1198781 119878119902

For each adjacent pair 119878120572 = [V(1205721) V(1205722) V(120572119896+1)]119878120573 = [V(1205731) V(1205732) V(120573119896+1)] of 119896-cells in Δ assume that119878120572 cap 119878120573 = [V(1205721) V(120572119894minus1) V(120572119894) V(120572119894+1) V(120572119896+1)] =

[V(1205731) V(120573119895minus1) V(120573119895) V(120573119895+1) V(120573119896+1)] where the hatmeans that the corresponding vertex is omitted Obvi-ously there exist two arrangements 1198971 1198972 119897119896+1 and 1198891 1198892 119889119896+1 of numbers 1 2 119896 + 1 such that V(120572119897

1) =

V(120572119894) V(1205731198891) = V(120573119895) and V(120572119897

119908) = V(120573119889

119908) for 119908 = 2 119896 + 1

Now we define two one-to-one transformations 120578120572120573120578120573120572 (R+)

119896+1rarr (R+)

119896+1 by 120578120572120573(120582) = (1205821198971

1205821198972

120582119897119896+1

)and 120578120573120572(120582) = (120582119889

1

1205821198892

120582119889119896+1

) respectively where 120582 =

(1205821 1205822 120582119896+1) isin (R+)119896+1

For any given 120572 isin 1 119902 let 119891(1205721) 119891

(120572119898120572) be BB-

polynomials over the simplex 119878120572 = [V(1205721) V(1205722) V(120572119896+1)]with119873(119891(120572120577)

) = 119875(120572120577) 120577 = 1 119898120572 and let

119891[120572]

119905(120591) = sum

|120582|=119899

119887[120572]

120582119861120582119899 (120591) 119905

][120572](120582119899)(18)

be a 119901119886119905119888ℎ119908119900119903119896119894119899119892 of BB-polynomials 119891(1205721) 119891

(120572119898120572) by

][120572]

Abstract and Applied Analysis 5

Define the piecewise BB-polynomial 119891119905 on the domainΩby

119891119905

1003816100381610038161003816119878120572

(120591) = 119891[120572]

119905(120591) 120572 = 1 2 119902 (19)

Theorem10 Suppose that119891(11) 119891

(11198981) 119891

(21) 119891

(119902119898119902)

are non-degenerate that 119891[120572]

119905is a 119901119886119905119888ℎ119908119900119903119896119894119899119892 of BB-

polynomials 119891(1205721) 119891

(120572119898120572) by ][120572] 120572 = 1 119902 and that

119878120572 119878120573 V(120572119894) V(120573119895) 120578120572120573 120578120573120572 119875(120572120577) 119891[120572]

119905 and 119891119905 are defined

as above and Int(119875(120572120577))⋂ Int(119875(120572120589)) = 0 for 120577 = 120589 If for eachadjacent pair 119878120572 119878120573 of 119896-cells in Δ and all 120588 isin 0 1 119903 and120582 = (1205821 1205822 120582119896+1) isin Z+(119896119899) with sum

119896+1

119894=2120582119894 = 119899 minus 120588

119887[120573]

(120578minus1

120573120572(120582120588))119905][120573](120578minus1

120573120572(120582120588)119899)

= sum

|120598|=120588

119887[120572]

120578minus1

120572120573(120598+1205820)119861120598120588 (120591

1015840

120573120572) 119905

][120572](120578minus1120572120573(120598+1205820)119899) 119905 isin R

lowast

+

(20)

where 120582120588 = (120588 1205822 120582119896+1) and 1205911015840120573120572 denotes the barycen-tric coordinate of V(120573119895) with respect to the simplex 119878120572 =

[V(1205721198971) V(120572119897

2) V(120572119897

119896+1)] then

(1) 119891119905 isin 119862119903

119899(Δ)

(2) if for any BB-polynomial 119891(120572119894) with 119873(119891(120572119894))⋂

(119878120572⋂119878120573) = 0 119894 isin 1 119898120572 there exists a BB-polynomial 119891(120573119895) in 119891

(1205731) 119891

(120573119898120573) such that

119873(119891(120573119895)

)⋂(119878120572⋂119878120573) = 119873(119891(120572119894))⋂(119878120572⋂119878120573) then

there is 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart119862119905(119891119905) of piecewise BB-polynomial 119891119905 up to isotopy is⋃

119902

120572=1⋃

119898120572

119894=1119862119905119878120572

(119891(120572119894))

Proof Let 120591 = (1205911 120591119896+1) and 120594 = (1205941 1205942 120594119896+1) bethe barycentric coordinates of point 119901 isin 119878120573 with respectto 119878120573 = [V(1205731) V(1205732) V(120573119896+1)] and 119878120573 = [V(120573119889

1) V(120573119889

2)

V(120573119889119896+1

)] respectively Then

120594 = (1205941 1205942 120594119896+1) = 120578120573120572 (120591) = (1205911198891

1205911198892

120591119889119896+1

) (21)

by the definition of the transformation 120578120573120572 and so

119891[120573]

119905(119901) = 119891

[120573]

119905 (120591)

= sum

|120582|=119899

119887[120573]

120582

119899

12058211205822 sdot sdot sdot 120582119896+1

times 1205911205821

11205911205822

2sdot sdot sdot 120591

120582119896+1

119896+1119905][120573](120582119899)

= sum

|120582|=119899

119887[120573]

120582

119899

1205821198891

1205821198891

sdot sdot sdot 120582119889119896+1

times (1205941)1205821198891(1205942)

1205821198892sdot sdot sdot (120594119896+1)

120582119889119896+1 119905

][120573](120582119899)

(22)

Set 120583 = (1205831 1205832 120583119896+1) = 120578120573120572(120582) Then 120582 = 120578minus1120573120572(120583) and

(1205831 1205832 120583119896+1) = (1205821198891

1205821198892

120582119889119896+1

) Therefore it followsfrom equality (22) that

119891[120573]

119905(119901) = sum

|120583|=119899

119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899)

times

119899

12058311205831 sdot sdot sdot 120583119896+1

(1205941)1205831

(1205942)1205832

sdot sdot sdot (120594119896+1)120583119896+1

(23)

This shows that the polynomial 119891[120573]

119905over 119878120573 = [V(120573119889

1) V(120573119889

2)

V(120573119889119896+1

)] can be represented as follows

119891[120573]

119905(120594) = sum

|120583|=119899

119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899)

119861120583119899 (120594) (24)

By a similar argument above we have that the polynomial119891[120572]

119905over 119878120572 = [V(120572119897

1) V(120572119897

119896+1)] can be represented as

follows

119891[120572]

119905(120594) = sum

|120583|=119899

119887[120572]

120578minus1

120572120573(120583)119905][120572](120578minus1

120572120573(120583)119899)

119861120583119899 (120594) (25)

where 120594 = (1205941 120594

2 120594

119896+1) is the barycentric coordinate of

point 119901 isin 119878120572 with respect to 119878120572 = [V(1205721198971) V(120572119897

2) V(120572119897

119896+1)]

Since V(120572119897119908) = V(120573119889

119908) for 119908 = 2 119896 + 1 119878120572⋂119878120573 =

[V(1205721198972) V(120572119897

119896+1)] thus according to Proposition 5 119891119905 isin

119862119903

119899(Δ) if and only if the BB-coefficients 119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899) of

119891[120573]

119905and the BB-coefficients 119887[120572]

120578minus1

120572120573(120583)119905][120572](120578minus1

120572120573(120583)119899) of 119891[120572]

119905satisfy

equality (20) This proves property (1)Below we will show that property (2) holdsFor each adjacent pair 119878120572 119878120573 of 119896-cells in Δ set Γ120572120573 =

119873(119891(120572119894))⋂(119878120572⋂119878120573)when119873(119891

(120572119894))⋂(119878120572⋂119878120573) = 0 then Γ120572120573

is a face of 119873(119891(120572119894)) and 119873(119891(120573119895)

) by the assumption that119873(119891

(120573119895))⋂(119878120572⋂119878120573) = 119873(119891

(120572119894))⋂(119878120572⋂119878120573) Since 119891[120572]

119905

(resp 119891[120573]

119905) is a patchworking of BB-polynomials 119891(1205721)

119891(120572119898120572) (resp 119891(1205731)

119891(120573119898120573)) by ][120572] (resp ][120573]) and

Int(119875(120572120577))⋂ Int(119875(120572120589)) = Int(119875(120573120577))⋂ Int(119875(120573120589)) = 0 for 120577 = 120589then for Γ120572120573-truncations (119891

(120572119894))Γ120572120573 (119891(120573119895)

)Γ120572120573 (119891[120572]

119905)Γ120572120573 and

(119891[120573]

119905)Γ120572120573 of 119891(120572119894) 119891(120573119895) 119891[120572]

119905 and 119891[120573]

119905 respectively we have

that

(119891(120572119894))

Γ120572120573

= (119891[120572]

119905)

Γ1205721205731003816100381610038161003816100381610038161003816119905=1

(119891(120573119895)

)

Γ120572120573

= (119891[120573]

119905)

Γ1205721205731003816100381610038161003816100381610038161003816119905=1

(26)

Since119891[120572]

119905and119891[120573]

119905are119862119903 smoothly connected on 119878120572⋂119878120573

119873(119891(120573119895)

)⋂(119878120572⋂119878120573) = 119873(119891(120572119894))⋂(119878120572⋂119878120573) and (20) implies

that (119891[120572]

119905)Γ120572120573= (119891

[120573]

119905)Γ120572120573 Thus (119891(120572119894)

)Γ120572120573= (119891

(120573119895))Γ120572120573 by (26)

6 Abstract and Applied Analysis

Using a similar argument in the proof of Theorem 6 wehave 119862119905119878

120572

(119891(120572119894))⋂ Γ120572120573 = 119862119905119878

120573

(119891(120573119895)

)⋂ Γ120572120573 by (119891(120572119894))Γ120572120573=

(119891(120573119895)

)Γ120572120573 This shows that

119898120572

⋃

119894=1

119862119905119878120572

(119891(120572119894))⋂(119878120572⋂119878120573)

=

119898120573

⋃

119895=1

119862119905119878120573

(119891(120573119895)

)⋂(119878120572⋂119878120573)

(27)

On the other hand since 119891[120572]

119905is a patchworking of

BB-polynomials 119891(1205721) 119891

(120572119898120572) by ][120572] we get from the

assumption and Proposition 3 that there is 119905[120572]0gt 0 such that

for any 119905 isin (0 119905[120572]0] the chart119862119905119878

120572

(119891[120572]

119905) of BB-polynomial119891[120572]

119905

(ie 119891119905|119878120572

) up to isotopy is⋃119898120572

119894=1119862119905119878120572

(119891(120572119894)) This conclusion

together with Theorems 7ndash9 and equality (27) implies thatthere is 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart119862119905(119891119905) of piecewise BB-polynomial 119891119905 up to isotopy is⋃

119902

120572=1⋃

119898120572

119894=1119862119905119878120572

(119891(120572119894)) where 1199050 = min119905[1]

0 119905

[119902]

0 This

completes the proof

Assume that 119875 is a 119896-dimensional simplex with domainpoints in 119878 as its vertices and that the BB-polynomial 119891 is areal 119896 + 1-nomial (this means that the nonzero coefficients in119891 correspond to the only vertices of 119875) Denote by119872(119875) theset of middle points of edges of 119875 For any point V isin 119872(119875) onan edge with endpoints (1198941119899 119894119896+1119899) (1198951119899 119895119896+1119899)we assign that

120575 (V) = sign (119887(1198941119894119896+1

) sdot 119887(1198951119895119896+1

)) (28)

where 119887(1198941119894119896+1

) 119887(1198951119895119896+1

) are the corresponding BB-coefficients in 119891 Put

119872minus (119875) = V isin 119872 (119875) | 120575 (V) = minus1 (29)

Theorem 11 Suppose that 119891(11) 119891

(11198981) 119891

(1199021)

119891(119902119898119902) 119875(11) 119875(1119898

1) 119875(1199021) 119875(119902119898

119902) 119891

[120572]

119905 and119891119905 sat-

isfy the conditions in Theorem 10 and that for each 120572 isin

1 2 119902 119891(1205721) 119891

(120572119898120572) are real (119896 + 1)-nomial and

119875(1205721) 119875(120572119898120572) with dimension 119896 form a triangulation of

119873(119891[120572]

119905) Then 119891119905 isin 119862119903

119899(Δ) and there exists a constant 1199050 gt 0

such that for any 119905 isin (0 1199050] the chart 119862119905(119891) of 119891 up to isotopyis ⋃119902

120572=1⋃

119898120572

119894=1conv(119872minus(119875(120572119894))) where 119872minus(119875(120572119894)) is defined as

(29)

Proof By assumption and [18 Theorem 44] we know thatthere is 119905[120572]

0gt 0 such that for any 119905 isin (0 119905

[120572]

0] the

chart 119862119905119878120572

(119891[120572]

119905) of BB-polynomial 119891[120572]

119905 up to isotopy is

⋃119898120572

119894=1conv(119872minus(119875(120572119894)))

The remainder of the proof can be completed by a similarapproach in the proof of Theorem 10

According toTheorems 10 and 11 if we want to constructa 119862119903 piecewise algebraic hypersurface 119891(120591) = 0 with aprescribed complex topology and a degree on a partition

Δ we can just construct some Bernstein-Bezier algebraichypersurfaces pieces 119891(1205721)

(120591) = 0 119891(120572119898120572)(120591) = 0 with

simple topology and a convex function ][120572] on each 119896-cell119878120572 where 119891

(1205721) 119891

(120572119898120572) and ][120572] satisfy the conditions

of Theorem 10 or Theorem 11 120572 = 1 2 119902 and then theBB-polynomial 119891[120572]

(120591) is obtained by a patchworking of BB-polynomials 119891(1205721)

119891(120572119898120572) by ][120572] Thus we can get 119891(120591)

by defining 119891|119878120572

(120591) = 119891[120572](120591)

Conflict of Interests

The authors do not have any possible conflict of interests inthis paper

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (Grant nos 11271328 11071031U0935004 U1135003 11290143 11101366 11226329 and61272307) the Zhejiang Provincial Natural Science Foun-dation (nos Y7080068 LQ13A010004) and the Foundationof Department of Education of Zhejiang Province (noY201223556)

References

[1] Y S Lai R H Wang and J M Wu ldquoReal zeros of the zero-dimensional parametric piecewise algebraic varietyrdquo Science inChina A vol 52 no 4 pp 817ndash832 2009

[2] Y S Lai R H Wang and J M Wu ldquoSolving parametricpiecewise polynomial systemsrdquo Journal of Computational andApplied Mathematics vol 236 no 5 pp 924ndash936 2011

[3] R-H Wang X Q Shi Z X Luo and Z X Su Multivariatespline and its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001

[4] O Ya Viro ldquoGluing of algebraic hypersurfaces smoothingof singularities and constructionof curvesrdquo in Proceedings ofthe Leningrad International Topological Conference pp 149ndash197Nauka Leningrad Russia 1983 (Russian)

[5] O Ya Viro ldquoGluing of plane real algebraic curves and construc-tions of curves of degrees 6 and 7rdquo inTopology (Leningrad 1982)vol 1060 of Lecture Notes inMathematics pp 187ndash200 SpringerBerlin Germany 1984

[6] O Ya Viro ldquoPatchworking real algebraic varietiesrdquo httpwwwmathsunysbedusimolegpwpdf

[7] E Shustin ldquoCritical points of real polynomials subdivisionsof Newton polyhedra and topology of real algebraic hypersur-facesrdquo Transactions of the American Mathematical Society vol173 no 2 pp 203ndash223 1996

[8] E Shustin ldquoGluing of singular and critical pointsrdquoTopology vol37 no 1 pp 195ndash217 1998

[9] E Shustin ldquoReal plane algebraic curves with prescribed singu-laritiesrdquo Topology vol 32 no 4 pp 845ndash856 1993

[10] F Bihan ldquoViro method for the construction of real completeintersectionsrdquo Advances in Mathematics vol 169 no 2 pp 177ndash186 2002

[11] J J Risler ldquoConstruction drsquohypersurfaces reelles [drsquoapres Viro]rdquoin Seminaire Bourbaki vol 199293 of Asterique no 216 Exp7633 pp 69ndash86 1993

Abstract and Applied Analysis 7

[12] I Itenberg and E Shustin ldquoSingular points and limit cycles ofplanar polynomial vector fieldsrdquo Duke Mathematical Journalvol 102 no 1 pp 1ndash37 2000

[13] I Itenberg and E Shustin ldquoViro theorem and topology of realand complex combinatorial hypersurfacesrdquo Israel Journal ofMathematics vol 133 no 1 pp 189ndash238 2003

[14] B Sturmfels ldquoVirorsquos theorem for complete intersectionsrdquoAnnalidella Scuola Normale Superiore di Pisa vol 21 no 3 pp 377ndash386 1994

[15] G Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Guide (Computer Science and ScientificComputing) Academic Press San Diego Calif USA 4thedition 1997

[16] C M Hoffmann Geometric and Solid Modeling Morgan Kauf-mann San Mateo Calif USA 1989

[17] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design AK Peters Boston Mass USA 1993

[18] Y S Lai W P Du and R H Wang ldquoThe Viro method forconstruction of Bernstein-Bezier algebraic hypersurface piecerdquoScience China Mathematics vol 55 no 6 pp 1269ndash1279 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Define the piecewise BB-polynomial 119891119905 on the domainΩby

119891119905

1003816100381610038161003816119878120572

(120591) = 119891[120572]

119905(120591) 120572 = 1 2 119902 (19)

Theorem10 Suppose that119891(11) 119891

(11198981) 119891

(21) 119891

(119902119898119902)

are non-degenerate that 119891[120572]

119905is a 119901119886119905119888ℎ119908119900119903119896119894119899119892 of BB-

polynomials 119891(1205721) 119891

(120572119898120572) by ][120572] 120572 = 1 119902 and that

119878120572 119878120573 V(120572119894) V(120573119895) 120578120572120573 120578120573120572 119875(120572120577) 119891[120572]

119905 and 119891119905 are defined

as above and Int(119875(120572120577))⋂ Int(119875(120572120589)) = 0 for 120577 = 120589 If for eachadjacent pair 119878120572 119878120573 of 119896-cells in Δ and all 120588 isin 0 1 119903 and120582 = (1205821 1205822 120582119896+1) isin Z+(119896119899) with sum

119896+1

119894=2120582119894 = 119899 minus 120588

119887[120573]

(120578minus1

120573120572(120582120588))119905][120573](120578minus1

120573120572(120582120588)119899)

= sum

|120598|=120588

119887[120572]

120578minus1

120572120573(120598+1205820)119861120598120588 (120591

1015840

120573120572) 119905

][120572](120578minus1120572120573(120598+1205820)119899) 119905 isin R

lowast

+

(20)

where 120582120588 = (120588 1205822 120582119896+1) and 1205911015840120573120572 denotes the barycen-tric coordinate of V(120573119895) with respect to the simplex 119878120572 =

[V(1205721198971) V(120572119897

2) V(120572119897

119896+1)] then

(1) 119891119905 isin 119862119903

119899(Δ)

(2) if for any BB-polynomial 119891(120572119894) with 119873(119891(120572119894))⋂

(119878120572⋂119878120573) = 0 119894 isin 1 119898120572 there exists a BB-polynomial 119891(120573119895) in 119891

(1205731) 119891

(120573119898120573) such that

119873(119891(120573119895)

)⋂(119878120572⋂119878120573) = 119873(119891(120572119894))⋂(119878120572⋂119878120573) then

there is 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart119862119905(119891119905) of piecewise BB-polynomial 119891119905 up to isotopy is⋃

119902

120572=1⋃

119898120572

119894=1119862119905119878120572

(119891(120572119894))

Proof Let 120591 = (1205911 120591119896+1) and 120594 = (1205941 1205942 120594119896+1) bethe barycentric coordinates of point 119901 isin 119878120573 with respectto 119878120573 = [V(1205731) V(1205732) V(120573119896+1)] and 119878120573 = [V(120573119889

1) V(120573119889

2)

V(120573119889119896+1

)] respectively Then

120594 = (1205941 1205942 120594119896+1) = 120578120573120572 (120591) = (1205911198891

1205911198892

120591119889119896+1

) (21)

by the definition of the transformation 120578120573120572 and so

119891[120573]

119905(119901) = 119891

[120573]

119905 (120591)

= sum

|120582|=119899

119887[120573]

120582

119899

12058211205822 sdot sdot sdot 120582119896+1

times 1205911205821

11205911205822

2sdot sdot sdot 120591

120582119896+1

119896+1119905][120573](120582119899)

= sum

|120582|=119899

119887[120573]

120582

119899

1205821198891

1205821198891

sdot sdot sdot 120582119889119896+1

times (1205941)1205821198891(1205942)

1205821198892sdot sdot sdot (120594119896+1)

120582119889119896+1 119905

][120573](120582119899)

(22)

Set 120583 = (1205831 1205832 120583119896+1) = 120578120573120572(120582) Then 120582 = 120578minus1120573120572(120583) and

(1205831 1205832 120583119896+1) = (1205821198891

1205821198892

120582119889119896+1

) Therefore it followsfrom equality (22) that

119891[120573]

119905(119901) = sum

|120583|=119899

119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899)

times

119899

12058311205831 sdot sdot sdot 120583119896+1

(1205941)1205831

(1205942)1205832

sdot sdot sdot (120594119896+1)120583119896+1

(23)

This shows that the polynomial 119891[120573]

119905over 119878120573 = [V(120573119889

1) V(120573119889

2)

V(120573119889119896+1

)] can be represented as follows

119891[120573]

119905(120594) = sum

|120583|=119899

119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899)

119861120583119899 (120594) (24)

By a similar argument above we have that the polynomial119891[120572]

119905over 119878120572 = [V(120572119897

1) V(120572119897

119896+1)] can be represented as

follows

119891[120572]

119905(120594) = sum

|120583|=119899

119887[120572]

120578minus1

120572120573(120583)119905][120572](120578minus1

120572120573(120583)119899)

119861120583119899 (120594) (25)

where 120594 = (1205941 120594

2 120594

119896+1) is the barycentric coordinate of

point 119901 isin 119878120572 with respect to 119878120572 = [V(1205721198971) V(120572119897

2) V(120572119897

119896+1)]

Since V(120572119897119908) = V(120573119889

119908) for 119908 = 2 119896 + 1 119878120572⋂119878120573 =

[V(1205721198972) V(120572119897

119896+1)] thus according to Proposition 5 119891119905 isin

119862119903

119899(Δ) if and only if the BB-coefficients 119887[120573]

120578minus1

120573120572(120583)119905][120573](120578minus1

120573120572(120583)119899) of

119891[120573]

119905and the BB-coefficients 119887[120572]

120578minus1

120572120573(120583)119905][120572](120578minus1

120572120573(120583)119899) of 119891[120572]

119905satisfy

equality (20) This proves property (1)Below we will show that property (2) holdsFor each adjacent pair 119878120572 119878120573 of 119896-cells in Δ set Γ120572120573 =

119873(119891(120572119894))⋂(119878120572⋂119878120573)when119873(119891

(120572119894))⋂(119878120572⋂119878120573) = 0 then Γ120572120573

is a face of 119873(119891(120572119894)) and 119873(119891(120573119895)

) by the assumption that119873(119891

(120573119895))⋂(119878120572⋂119878120573) = 119873(119891

(120572119894))⋂(119878120572⋂119878120573) Since 119891[120572]

119905

(resp 119891[120573]

119905) is a patchworking of BB-polynomials 119891(1205721)

119891(120572119898120572) (resp 119891(1205731)

119891(120573119898120573)) by ][120572] (resp ][120573]) and

Int(119875(120572120577))⋂ Int(119875(120572120589)) = Int(119875(120573120577))⋂ Int(119875(120573120589)) = 0 for 120577 = 120589then for Γ120572120573-truncations (119891

(120572119894))Γ120572120573 (119891(120573119895)

)Γ120572120573 (119891[120572]

119905)Γ120572120573 and

(119891[120573]

119905)Γ120572120573 of 119891(120572119894) 119891(120573119895) 119891[120572]

119905 and 119891[120573]

119905 respectively we have

that

(119891(120572119894))

Γ120572120573

= (119891[120572]

119905)

Γ1205721205731003816100381610038161003816100381610038161003816119905=1

(119891(120573119895)

)

Γ120572120573

= (119891[120573]

119905)

Γ1205721205731003816100381610038161003816100381610038161003816119905=1

(26)

Since119891[120572]

119905and119891[120573]

119905are119862119903 smoothly connected on 119878120572⋂119878120573

119873(119891(120573119895)

)⋂(119878120572⋂119878120573) = 119873(119891(120572119894))⋂(119878120572⋂119878120573) and (20) implies

that (119891[120572]

119905)Γ120572120573= (119891

[120573]

119905)Γ120572120573 Thus (119891(120572119894)

)Γ120572120573= (119891

(120573119895))Γ120572120573 by (26)

6 Abstract and Applied Analysis

Using a similar argument in the proof of Theorem 6 wehave 119862119905119878

120572

(119891(120572119894))⋂ Γ120572120573 = 119862119905119878

120573

(119891(120573119895)

)⋂ Γ120572120573 by (119891(120572119894))Γ120572120573=

(119891(120573119895)

)Γ120572120573 This shows that

119898120572

⋃

119894=1

119862119905119878120572

(119891(120572119894))⋂(119878120572⋂119878120573)

=

119898120573

⋃

119895=1

119862119905119878120573

(119891(120573119895)

)⋂(119878120572⋂119878120573)

(27)

On the other hand since 119891[120572]

119905is a patchworking of

BB-polynomials 119891(1205721) 119891

(120572119898120572) by ][120572] we get from the

assumption and Proposition 3 that there is 119905[120572]0gt 0 such that

for any 119905 isin (0 119905[120572]0] the chart119862119905119878

120572

(119891[120572]

119905) of BB-polynomial119891[120572]

119905

(ie 119891119905|119878120572

) up to isotopy is⋃119898120572

119894=1119862119905119878120572

(119891(120572119894)) This conclusion

together with Theorems 7ndash9 and equality (27) implies thatthere is 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart119862119905(119891119905) of piecewise BB-polynomial 119891119905 up to isotopy is⋃

119902

120572=1⋃

119898120572

119894=1119862119905119878120572

(119891(120572119894)) where 1199050 = min119905[1]

0 119905

[119902]

0 This

completes the proof

Assume that 119875 is a 119896-dimensional simplex with domainpoints in 119878 as its vertices and that the BB-polynomial 119891 is areal 119896 + 1-nomial (this means that the nonzero coefficients in119891 correspond to the only vertices of 119875) Denote by119872(119875) theset of middle points of edges of 119875 For any point V isin 119872(119875) onan edge with endpoints (1198941119899 119894119896+1119899) (1198951119899 119895119896+1119899)we assign that

120575 (V) = sign (119887(1198941119894119896+1

) sdot 119887(1198951119895119896+1

)) (28)

where 119887(1198941119894119896+1

) 119887(1198951119895119896+1

) are the corresponding BB-coefficients in 119891 Put

119872minus (119875) = V isin 119872 (119875) | 120575 (V) = minus1 (29)

Theorem 11 Suppose that 119891(11) 119891

(11198981) 119891

(1199021)

119891(119902119898119902) 119875(11) 119875(1119898

1) 119875(1199021) 119875(119902119898

119902) 119891

[120572]

119905 and119891119905 sat-

isfy the conditions in Theorem 10 and that for each 120572 isin

1 2 119902 119891(1205721) 119891

(120572119898120572) are real (119896 + 1)-nomial and

119875(1205721) 119875(120572119898120572) with dimension 119896 form a triangulation of

119873(119891[120572]

119905) Then 119891119905 isin 119862119903

119899(Δ) and there exists a constant 1199050 gt 0

such that for any 119905 isin (0 1199050] the chart 119862119905(119891) of 119891 up to isotopyis ⋃119902

120572=1⋃

119898120572

119894=1conv(119872minus(119875(120572119894))) where 119872minus(119875(120572119894)) is defined as

(29)

Proof By assumption and [18 Theorem 44] we know thatthere is 119905[120572]

0gt 0 such that for any 119905 isin (0 119905

[120572]

0] the

chart 119862119905119878120572

(119891[120572]

119905) of BB-polynomial 119891[120572]

119905 up to isotopy is

⋃119898120572

119894=1conv(119872minus(119875(120572119894)))

The remainder of the proof can be completed by a similarapproach in the proof of Theorem 10

According toTheorems 10 and 11 if we want to constructa 119862119903 piecewise algebraic hypersurface 119891(120591) = 0 with aprescribed complex topology and a degree on a partition

Δ we can just construct some Bernstein-Bezier algebraichypersurfaces pieces 119891(1205721)

(120591) = 0 119891(120572119898120572)(120591) = 0 with

simple topology and a convex function ][120572] on each 119896-cell119878120572 where 119891

(1205721) 119891

(120572119898120572) and ][120572] satisfy the conditions

of Theorem 10 or Theorem 11 120572 = 1 2 119902 and then theBB-polynomial 119891[120572]

(120591) is obtained by a patchworking of BB-polynomials 119891(1205721)

119891(120572119898120572) by ][120572] Thus we can get 119891(120591)

by defining 119891|119878120572

(120591) = 119891[120572](120591)

Conflict of Interests

The authors do not have any possible conflict of interests inthis paper

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (Grant nos 11271328 11071031U0935004 U1135003 11290143 11101366 11226329 and61272307) the Zhejiang Provincial Natural Science Foun-dation (nos Y7080068 LQ13A010004) and the Foundationof Department of Education of Zhejiang Province (noY201223556)

References

[1] Y S Lai R H Wang and J M Wu ldquoReal zeros of the zero-dimensional parametric piecewise algebraic varietyrdquo Science inChina A vol 52 no 4 pp 817ndash832 2009

[2] Y S Lai R H Wang and J M Wu ldquoSolving parametricpiecewise polynomial systemsrdquo Journal of Computational andApplied Mathematics vol 236 no 5 pp 924ndash936 2011

[3] R-H Wang X Q Shi Z X Luo and Z X Su Multivariatespline and its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001

[4] O Ya Viro ldquoGluing of algebraic hypersurfaces smoothingof singularities and constructionof curvesrdquo in Proceedings ofthe Leningrad International Topological Conference pp 149ndash197Nauka Leningrad Russia 1983 (Russian)

[5] O Ya Viro ldquoGluing of plane real algebraic curves and construc-tions of curves of degrees 6 and 7rdquo inTopology (Leningrad 1982)vol 1060 of Lecture Notes inMathematics pp 187ndash200 SpringerBerlin Germany 1984

[6] O Ya Viro ldquoPatchworking real algebraic varietiesrdquo httpwwwmathsunysbedusimolegpwpdf

[7] E Shustin ldquoCritical points of real polynomials subdivisionsof Newton polyhedra and topology of real algebraic hypersur-facesrdquo Transactions of the American Mathematical Society vol173 no 2 pp 203ndash223 1996

[8] E Shustin ldquoGluing of singular and critical pointsrdquoTopology vol37 no 1 pp 195ndash217 1998

[9] E Shustin ldquoReal plane algebraic curves with prescribed singu-laritiesrdquo Topology vol 32 no 4 pp 845ndash856 1993

[10] F Bihan ldquoViro method for the construction of real completeintersectionsrdquo Advances in Mathematics vol 169 no 2 pp 177ndash186 2002

[11] J J Risler ldquoConstruction drsquohypersurfaces reelles [drsquoapres Viro]rdquoin Seminaire Bourbaki vol 199293 of Asterique no 216 Exp7633 pp 69ndash86 1993

Abstract and Applied Analysis 7

[12] I Itenberg and E Shustin ldquoSingular points and limit cycles ofplanar polynomial vector fieldsrdquo Duke Mathematical Journalvol 102 no 1 pp 1ndash37 2000

[13] I Itenberg and E Shustin ldquoViro theorem and topology of realand complex combinatorial hypersurfacesrdquo Israel Journal ofMathematics vol 133 no 1 pp 189ndash238 2003

[14] B Sturmfels ldquoVirorsquos theorem for complete intersectionsrdquoAnnalidella Scuola Normale Superiore di Pisa vol 21 no 3 pp 377ndash386 1994

[15] G Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Guide (Computer Science and ScientificComputing) Academic Press San Diego Calif USA 4thedition 1997

[16] C M Hoffmann Geometric and Solid Modeling Morgan Kauf-mann San Mateo Calif USA 1989

[17] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design AK Peters Boston Mass USA 1993

[18] Y S Lai W P Du and R H Wang ldquoThe Viro method forconstruction of Bernstein-Bezier algebraic hypersurface piecerdquoScience China Mathematics vol 55 no 6 pp 1269ndash1279 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Using a similar argument in the proof of Theorem 6 wehave 119862119905119878

120572

(119891(120572119894))⋂ Γ120572120573 = 119862119905119878

120573

(119891(120573119895)

)⋂ Γ120572120573 by (119891(120572119894))Γ120572120573=

(119891(120573119895)

)Γ120572120573 This shows that

119898120572

⋃

119894=1

119862119905119878120572

(119891(120572119894))⋂(119878120572⋂119878120573)

=

119898120573

⋃

119895=1

119862119905119878120573

(119891(120573119895)

)⋂(119878120572⋂119878120573)

(27)

On the other hand since 119891[120572]

119905is a patchworking of

BB-polynomials 119891(1205721) 119891

(120572119898120572) by ][120572] we get from the

assumption and Proposition 3 that there is 119905[120572]0gt 0 such that

for any 119905 isin (0 119905[120572]0] the chart119862119905119878

120572

(119891[120572]

119905) of BB-polynomial119891[120572]

119905

(ie 119891119905|119878120572

) up to isotopy is⋃119898120572

119894=1119862119905119878120572

(119891(120572119894)) This conclusion

together with Theorems 7ndash9 and equality (27) implies thatthere is 1199050 gt 0 such that for any 119905 isin (0 1199050] the chart119862119905(119891119905) of piecewise BB-polynomial 119891119905 up to isotopy is⋃

119902

120572=1⋃

119898120572

119894=1119862119905119878120572

(119891(120572119894)) where 1199050 = min119905[1]

0 119905

[119902]

0 This

completes the proof

Assume that 119875 is a 119896-dimensional simplex with domainpoints in 119878 as its vertices and that the BB-polynomial 119891 is areal 119896 + 1-nomial (this means that the nonzero coefficients in119891 correspond to the only vertices of 119875) Denote by119872(119875) theset of middle points of edges of 119875 For any point V isin 119872(119875) onan edge with endpoints (1198941119899 119894119896+1119899) (1198951119899 119895119896+1119899)we assign that

120575 (V) = sign (119887(1198941119894119896+1

) sdot 119887(1198951119895119896+1

)) (28)

where 119887(1198941119894119896+1

) 119887(1198951119895119896+1

) are the corresponding BB-coefficients in 119891 Put

119872minus (119875) = V isin 119872 (119875) | 120575 (V) = minus1 (29)

Theorem 11 Suppose that 119891(11) 119891

(11198981) 119891

(1199021)

119891(119902119898119902) 119875(11) 119875(1119898

1) 119875(1199021) 119875(119902119898

119902) 119891

[120572]

119905 and119891119905 sat-

isfy the conditions in Theorem 10 and that for each 120572 isin

1 2 119902 119891(1205721) 119891

(120572119898120572) are real (119896 + 1)-nomial and

119875(1205721) 119875(120572119898120572) with dimension 119896 form a triangulation of

119873(119891[120572]

119905) Then 119891119905 isin 119862119903

119899(Δ) and there exists a constant 1199050 gt 0

such that for any 119905 isin (0 1199050] the chart 119862119905(119891) of 119891 up to isotopyis ⋃119902

120572=1⋃

119898120572

119894=1conv(119872minus(119875(120572119894))) where 119872minus(119875(120572119894)) is defined as

(29)

Proof By assumption and [18 Theorem 44] we know thatthere is 119905[120572]

0gt 0 such that for any 119905 isin (0 119905

[120572]

0] the

chart 119862119905119878120572

(119891[120572]

119905) of BB-polynomial 119891[120572]

119905 up to isotopy is

⋃119898120572

119894=1conv(119872minus(119875(120572119894)))

The remainder of the proof can be completed by a similarapproach in the proof of Theorem 10

According toTheorems 10 and 11 if we want to constructa 119862119903 piecewise algebraic hypersurface 119891(120591) = 0 with aprescribed complex topology and a degree on a partition

Δ we can just construct some Bernstein-Bezier algebraichypersurfaces pieces 119891(1205721)

(120591) = 0 119891(120572119898120572)(120591) = 0 with

simple topology and a convex function ][120572] on each 119896-cell119878120572 where 119891

(1205721) 119891

(120572119898120572) and ][120572] satisfy the conditions

of Theorem 10 or Theorem 11 120572 = 1 2 119902 and then theBB-polynomial 119891[120572]

(120591) is obtained by a patchworking of BB-polynomials 119891(1205721)

119891(120572119898120572) by ][120572] Thus we can get 119891(120591)

by defining 119891|119878120572

(120591) = 119891[120572](120591)

Conflict of Interests

The authors do not have any possible conflict of interests inthis paper

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (Grant nos 11271328 11071031U0935004 U1135003 11290143 11101366 11226329 and61272307) the Zhejiang Provincial Natural Science Foun-dation (nos Y7080068 LQ13A010004) and the Foundationof Department of Education of Zhejiang Province (noY201223556)

References

[1] Y S Lai R H Wang and J M Wu ldquoReal zeros of the zero-dimensional parametric piecewise algebraic varietyrdquo Science inChina A vol 52 no 4 pp 817ndash832 2009

[2] Y S Lai R H Wang and J M Wu ldquoSolving parametricpiecewise polynomial systemsrdquo Journal of Computational andApplied Mathematics vol 236 no 5 pp 924ndash936 2011

[3] R-H Wang X Q Shi Z X Luo and Z X Su Multivariatespline and its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001

[4] O Ya Viro ldquoGluing of algebraic hypersurfaces smoothingof singularities and constructionof curvesrdquo in Proceedings ofthe Leningrad International Topological Conference pp 149ndash197Nauka Leningrad Russia 1983 (Russian)

[5] O Ya Viro ldquoGluing of plane real algebraic curves and construc-tions of curves of degrees 6 and 7rdquo inTopology (Leningrad 1982)vol 1060 of Lecture Notes inMathematics pp 187ndash200 SpringerBerlin Germany 1984

[6] O Ya Viro ldquoPatchworking real algebraic varietiesrdquo httpwwwmathsunysbedusimolegpwpdf

[7] E Shustin ldquoCritical points of real polynomials subdivisionsof Newton polyhedra and topology of real algebraic hypersur-facesrdquo Transactions of the American Mathematical Society vol173 no 2 pp 203ndash223 1996

[8] E Shustin ldquoGluing of singular and critical pointsrdquoTopology vol37 no 1 pp 195ndash217 1998

[9] E Shustin ldquoReal plane algebraic curves with prescribed singu-laritiesrdquo Topology vol 32 no 4 pp 845ndash856 1993

[10] F Bihan ldquoViro method for the construction of real completeintersectionsrdquo Advances in Mathematics vol 169 no 2 pp 177ndash186 2002

[11] J J Risler ldquoConstruction drsquohypersurfaces reelles [drsquoapres Viro]rdquoin Seminaire Bourbaki vol 199293 of Asterique no 216 Exp7633 pp 69ndash86 1993

Abstract and Applied Analysis 7

[12] I Itenberg and E Shustin ldquoSingular points and limit cycles ofplanar polynomial vector fieldsrdquo Duke Mathematical Journalvol 102 no 1 pp 1ndash37 2000

[13] I Itenberg and E Shustin ldquoViro theorem and topology of realand complex combinatorial hypersurfacesrdquo Israel Journal ofMathematics vol 133 no 1 pp 189ndash238 2003

[14] B Sturmfels ldquoVirorsquos theorem for complete intersectionsrdquoAnnalidella Scuola Normale Superiore di Pisa vol 21 no 3 pp 377ndash386 1994

[15] G Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Guide (Computer Science and ScientificComputing) Academic Press San Diego Calif USA 4thedition 1997

[16] C M Hoffmann Geometric and Solid Modeling Morgan Kauf-mann San Mateo Calif USA 1989

[17] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design AK Peters Boston Mass USA 1993

[18] Y S Lai W P Du and R H Wang ldquoThe Viro method forconstruction of Bernstein-Bezier algebraic hypersurface piecerdquoScience China Mathematics vol 55 no 6 pp 1269ndash1279 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

[12] I Itenberg and E Shustin ldquoSingular points and limit cycles ofplanar polynomial vector fieldsrdquo Duke Mathematical Journalvol 102 no 1 pp 1ndash37 2000

[13] I Itenberg and E Shustin ldquoViro theorem and topology of realand complex combinatorial hypersurfacesrdquo Israel Journal ofMathematics vol 133 no 1 pp 189ndash238 2003

[14] B Sturmfels ldquoVirorsquos theorem for complete intersectionsrdquoAnnalidella Scuola Normale Superiore di Pisa vol 21 no 3 pp 377ndash386 1994

[15] G Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Guide (Computer Science and ScientificComputing) Academic Press San Diego Calif USA 4thedition 1997

[16] C M Hoffmann Geometric and Solid Modeling Morgan Kauf-mann San Mateo Calif USA 1989

[17] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design AK Peters Boston Mass USA 1993

[18] Y S Lai W P Du and R H Wang ldquoThe Viro method forconstruction of Bernstein-Bezier algebraic hypersurface piecerdquoScience China Mathematics vol 55 no 6 pp 1269ndash1279 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of