UNIVERSITI PUTRA MALAYSIA

ON THE ESTIMATE TO SOLUTIONS OF CONGRUENCE EQUATIONS ASSOCIATED WITH A QUARTIC FORM

CHAN KAIT LOON

FSAS 1997 6

ON THE ESTIMATE TO SOLUTIONS OF CONGRUENCE EQUATIONS ASSOCIATED WITH A QUARTIC FORM

BY

CHAN KAIT LOON

Thesis Submitted in Fulfilment of the Requirements for the degree of Master of Science in the

Faculty of Science and Environmental Studies Universiti Putra Malaysia

April 1997

ACKNOWLEDGEMENTS

I would like to express my most gratitude and sincere appreciation

to my chairman Prof. Dr. Kamel Ariffin bin Mohd. Atan and Dr. Ismail

Abdullah for their untiring guidance, valuable advice, support and comments.

Their patience and persistent encouragement during the course of my

research is instrumental to the completion of this thesis.

I also wish to thank the Head of Department, Assoc. Prof. Dr.

Harun Budin, academic and general staff of the Department of Mathematics,

Universiti Putra Malaysia, for their assistance in various capacities.

Last but not least, I would like to thank my family and friends for

their understanding, support and encouragement throughout the course of this

study .

i i

T ABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ........................................ ............... 11 LIST OF TABLE ........... . . ....................................................... . . v LIST OF FIGURES ...................................................... ...... ........ VI LIST OF SYMBOLS AND ABBREVIATIONS ....... . .......... .... x ABSTRACT . . . .. . ........ ... .... ....... ............. ....... ..... ...... ...... .... ............ Xll ABSTRAK .... . . . . .. . . ....... ..... . . ..... . . . . . ........ . . .. . . .. . . . ......................... . . . XIV

CHAPTER

I INTRODUCTION . . . . . . . . . . .. . .. . . .. . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . .

Notation and Definition ... ............ .. ..... .. .. . .. .. .. . . .. . . .. ... . .

Background . .. . . . .. . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . .. . .. . . . . . . . .. .. . . . . . . .

Organization of The Study . . .. .. . ... . ... . . .. . . . . . . . . . . . . . . . . . . . . .. . .

I I POL YNOMIALS AND NEWTON POLYGONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . Polynomial Rings . . . . . . . . . .. . . .... . . . . . . .. .. .. .. . .. . . . ... . . . . .. . . . ... . . . Roots of Polynomials .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .

Newton Polygons .... .. .. .. . .. .. .. . .. . . .. . ... . . . . . . . . . . . . . . .. .. .. ... . . ..

III NEWTON POLYHEDRONS AND INDICATOR DIAGRAMS . . . . . . . . . . . . . . . . .. . . . . . .. . . . .. . . . . . . .

Newton Polyhedrons . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. .. . .. .. . . . . . . . . ..

Normal to Newton Polyhedron .. .. ..... ... .... ...... . ..... .. . .. .

Indicator Diagrams ....... . . .. . . .. ... .. ..... .... . .... . .... ... . . . .... ... .

III

1 1 4 9

1 8 1 8 2 1 24

30 30 39 42

IV

V

VI

INTERSECTION OF INDICATOR DIAGRAMS . . . . . . .. ... . . . . . . . .. . .. . . .. . . . . . .. . . . . 52 p-adic Orders of Zeros for A Polynomial . . .. ..... ................ ............ ....... .................. 52 Common Zeros and Intersection of Indicator Diagrams . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . 54 Simple Intersection of Indicator Diagrams . . ....... . .. .... ...... ................ .......... ..... 63 Intersection with Coinciding Segments ...... .... . .. . . .... ........... ....... .................... ...... . . .. . 67

SET OF SOLUTIONS TO CONGRUENCE EQUATIONS ASSOCIATED WITH A QUARTIC FORM .. . . . . . . . . . . . . . . .. . . . . . .. .. .. . . .. . .. . . . .. .. . . . . .

Exponential Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .

p-adic Sizes of Common Zeros of fx and fy . . . . . . . . .. . . . . . .. . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . ..

Estimation ofN(fx , fy ; pU) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

Estimate for Exponential Sums . . . . . . . . . . . . . . . . . . . . . . . .. .. . . . . . . ..

CONCLUSION AND SUGGESTIONS ............... .. .. Major Findings . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suggestions for Further Research . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . ..

77 77

81 95 98

104 104 107 107

BIBLIOGRAPHY ........ . . ....... ............. ..... ... .. . . . ......... .... ... ............ ... 109

VITA . ....................... ............ ............................................................ 112

IV

LIST OF TABLE

Table Page

Maximum faces formed by a polynomial. .. ....... .... .. . . .. .. .. . .. . .. .. .. . .. . . . .. .... . . . .. .. . .. . . . .. . 36

v

Figure

1.

2.

3.

4

5.

6.

7.

8.

9.

LIST OF FIGURES

Newton polygon of f(x) = 3x4 - 4x3 - 26x2 -I- 36x - 9 with p = 3 ......... . . . . . . . . . .. . . . . .. . .. ............. ...... .. . .. . .. . .. . . .. . .. . . .. .

Newton polygon of f(x) = 1 12 X4 -I- 1 /2 x3 - 8x2 - 2x + 24 with p = 2 ...................................................................... .

Newton diagram of f(x,y) = 3x2 -I- 2xy _ y2 + 27 with p = 3 ..................................................................... . .

Newton diagram of f(x,y) = 8 -I- 4x2 -I- xy + 4xl with p = 2 ..................................................................... ..

Newton diagram of f(x,y) = 16 + 8x + 2y -I- 2x2y + y2 with p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . .

The Newton polyhedron of f(x,y) = 2 + x + xy with p = 2 ....................................... ..

The Newton polyhedron of f(x,y)=3x2 + 2xy -l + 27 with p = 3 ........................... . .

The Newton polyhedron of f(x,y)=8 + 4x2 + xy + 4xl with p = 2 ......................... . .

The Newton polyhedron of f(x,y) = 9x4 + 3x3y + xy2 + 9xy + 3y4 -I- 27 with p = 3 ............................................ . .

VI

Page

26

26

31

31

32

33

34

34

35

10.

11.

12.

13.

14.

15.

16.

17.

18.

1 9.

The Newton polyhedron of f(x,y) = 3X3y4 +- x4y2 + 27x4y + 9x2y + xy + 9x5 + 27x2y4 + 18xy5 + 9y + 9 with p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The projection Lr associated with the Nr off(x,y)=2+x+xywith p=2 ............................. .

The projection Lr associated with the Nf off(x,y) = 8 + 4x2 + xy + 4xy2 with p = 2 .................................................................... . .

Indicator diagram associated with the polynomial f(x,y) = 2 + x + xy with p = 2 ..................................................................... .

Indicator diagram associated with the polynomial f(x,y) = 3x2 + 2xy - y2 + 27

with p = 3 ..................................................................... .

Indicator diagram associated with the polynomial f(x,y) = 8 + 4x2 + xy + 4xl with p = 2 .................................................................... . .

Indicator diagram associated with the polynomial f(x,y)=9x4 + 3x3y + xy2 + 9xy + 3y4 + 27 with p = 3 ....................................................... .

Indicator diagram associated with the polynomial f(x,y) = 3x3l + x4y2 + 27x4y + 9x2y + xy + 9x5 + 27x2y4 + 18xi + 9y + 9 with P = 3 ............................................. .

The Indicator diagrams of f (x,y) = 2x + Y - 6 and g (x,y) = 4x + 3y - 16 with P = 3 ................................... .

The Indicator diagrams of f(x,y) = 2x + Y - 10 and g(x,y) = x + Y - 8 with p = 2 . ... . . . .... . . ... . .. . . . . . .. . . . . . . . ... . . . . . .

Vll

35

43

43

46

46

47

48

49

56

57

20. The Indicator diagrams of f(x,y) = 2X2 + 3xy + 3y2 + 8 and g(x,y) = x2 + 3xy + 4 with p = 2 ................................... . 58

21. The Indicator diagrams of f(x,y) = 1 + 3x + 4y and g(x,y) = 3 + x + 2y with p = 5 ........................ ............... 59

22. The Indicator diagrams of f(x,y) = 3x + 3y - 2 and g(x,y) = 2x + 2y - 4 with p = 3 ........ ............... ......... ..... 60

23. The Indicator diagrams of f(x,y) = 1 + 2x + 4y and g(x,y) = 3 + x + 2y with p = 5 ...................................... 61

24. The Indicator diagrams of f(x,y) = 3x + 3y + 1 and g(x,y) = x + Y + 2 with p = 2 . . . .... .................. ...... .... . . . . . 62

25. The Indicator diagrams of f(x,y) = 1 + 2x + 3y and g(x,y) = 1 + 3x + 5y with p = 3 ............................. .. ..... 65

26. The Indicator diagrams of f(x,y) = x + 5y - 6 and g(x,y) = xy + 5x - 26 with p = 5 ....... ...... . . . . . . . . .............. 66

27. The Indicator diagrams of f(x,y) = 2x + Y - 6 and g(x,y) = x + 2y - 10 with p = 3 ..................................... 68

28. The Indicator diagrams of f(x,y) = 5 + x + Sy and g(x,y) = 5 + x + xy with p = 5 ...................................... 70

29. The Indicator diagrams of f(x,y) = 7 + 3x + 2y and g(x,y) = 3 + 9x + 6y with p = 3 .................................... 71

VIIl

30.

31.

32.

The Indicator diagrams of f(x,y) = 2X2 + 3y - 5 and g(x,y) = 3x2 + 7y - 10 with p = 5 ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Indicator diagrams of f(x,y) = 12x2 + 4xy + y2 + 27 and g(x,y) = 2x2+ 2xy + 9 with p = 3 ........... ............. . . .... ... .

The Indicator diagrams of F(u,v) and G(u,v) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .

IX

72

73

75

p

ex

z

Q

R

C

F

F[�J

f Deg (f)

ordp a

Vf

Nr

V

E

L

LIST OF SYMBOLS AND ABBREVIATIONS

Prime Number

Exponent of Prime Numbers

Ring of Integers

Field of Rational Numbers

Field of Real Numbers

Field of Complex Numbers

Completion of Q p

n Tuple of Variable (XI, ... ,Xn)

Ring or Field

Ring of Polynomials with Coefficients in F

n Tuple of Polynomials (fJ , .. .,fm), m> 1

Degree off

Highest Power of p which Divides a

Gradient of f

Newton Polyhedron of f

Vertex ofNr

Edge ofNr

Projection ofNr

x

8 Determinant Factor

max Maximum

mll1 Minimum

mod Modulo

Exp Exponential

lip Valuation respect to p

I Summation

detA Determinant A

(a,b) Greatest Common Divisor of a and b

XI

Abstract of thesis submitted to the Senate of Universiti Putra Malaysia in fulfilment of the requirements for the degree of Master of Science.

ON THE ESTIMATE TO SOLUTIONS OF CONGRUENCE EQUATIONS ASSOCIATED WITH A QUARTIC FORM

By

CHAN KAIT LOON

APRIL 1997

Chairman : Professor Dr. Kamel Ariffin bin Mohd. Atan

Faculty : Science and Environmental Studies

The set of solutions to congruence equations modulo a prime power

associated with the polynomial

in Zp[x,y] is examined and its cardinality is estimated by employing the

Newton polyhedral technique.

XlI

The method involves reduction of the partial derivatives of f that is f"

and fy into polynomials with single variable and finding 8 the determinant

factor in the estimation. fx and fy are reduced to one-variable polynomials by

employment of suitable parameters. The Newton polyhedrons associated with

the polynomials so obtained are then considered and combination of their

Indicator diagrams examined.

There exist common zeros of the single- variable polynomials whose

p-adic orders correspond to the intersection points in the combination of the

Indicator diagrams associated with the respective Newton polyhedrons of the

polynomials. The p-adic sizes of these zeros are then determined, and this

leads to sizes of common zeros of the partial derivatives of f. This information

is then used to arrive at the estimate of the cardinality above.

X III

Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia sebagai memenuhi syarat bagi ljazah Master Sains.

PENGANGGARAN KEPADA PENYELESAIAN BAG I PERSAMAAN KONGRUEN BERSEKUTU DENGAN SATU BENTUK KUARTIK

Olch

CHAN KAIT LOON

APRIL 1 997

Pengerusi : Professor Dr. Kamel Ariffin bin Mohd. Atan

Fakulti : Sains dan Pengajian Alam Sekitar

Pcnganggaran kekardinalan kepada set penyelesaian bagi persamaan

kongruen modulo suatu kuasa perdana yang disekutukan dengan polinomial

dalam Zp [x,y] ditentukan dengan menggunakan teknik polihedron Newton.

XIV

Kaedah ini l11eliputi penurunan terbitan separa bagi f iaitu fx and fy

kepada polinol11ial satu pel11bolehubah, kel11udian faktor penentu (5 bagi

anggaran di atas diperolehi. Dalal11 proses penurunan fx and fy, paral11eter­

parameter yang sesuai digunakan. Kemudian polihedron Newton yang

disekutllkan dengan polinomial satu pembolehubah dipertimbangkan.

Terdapat pensifar-pensifar sepllnya dengan peringkat p-adic yang

bersepadanan dengan titik persilangan di dalam gabungan gambarajah

penllnjllk polihedron Newton polinol11ial-polinol11iaI ini. lni akan

menghasilkan saiz pensifar-pensifar sepunya bagi fx dan fy. Maklumat 1111

kemlldiannya digunakan untllk l11emperoleh anggaran kekardinalan di atas.

xv

CHAPTER I

INTRODUCTION

Notation and Definition

As usual, we use the standard notation Z, Q, R and C to denote ring of

integers, field of rational numbers, field of real numbers and field of

complex numbers respectively. With p denoting a prime number, Zp will

denote the ring of p-adic integers, Qp the field of p-adic numbers and Op the

completion of the algebraic closure of Qp.

The lower case of Roman letters will represent elements in Z or Zp and

the Greek letter a always denotes the exponent of a prime p.

With X denoting n tuple of variable (xt, . . . ,xn), n = 1 , 2, 3 , .. . , and F either a

ring or field, F[X] will mean the ring of polynomials with coefficients in F . In our

discussion F is either Z or Qp or field extensions of Qp

1

2

Let f = (fl ""'�n ) be m tuple of linear polynomials in F [�J . If f; = 2.:aijxj'

1::; i::; m, 1 ::; j ::; n, we call the m x n matrix [aij]' the matrix representing f and Jf

h·

J b' . [£fi] t e aco Ian matnx &j .

Suppose f = "ai 0 0 0i X/ I 0 0 ° Xn in is a polynomial in F [�]. The degree L....J I II

of f wil l be denoted by

deg(f) = max (il + ... + U· il .. ·i"

Let p be any prime number. For any nonzero integer a, ordp a will be the

highest power of p which divides a, that is the greatest a such that a == 0 (mod

If x = alb is any rational number, we define or� x to be ordp a - or� b.

This resembles the property of logarithm.

Further define a map lip on Q as fol lows:

3

1 if x:;t:O ordpx Ixlp = P

0 if X=O

It can be shown that lip is a non-Archimedean valuation on Q (Koblitz 1 977).

A sequence {aj} of rational numbers is called Cauchy sequence if given f:

> 0, there exists an N such that I ai - a., Ip < f:, for i, ( > N. Two Cauchy 1

sequences {ai} and {bi} are equivalent if

lim lai-bdp=O i�CX)

We define the field Qp to be the set of equivalence classes of Cauchy

sequences in Q, so Qp is the completion of Q with respect to lip

We denote by Q p' the algebraically closed field of Qp and Dp the

completion of Q p with respect to lip. It is found that the process of extending lip

from Qp to Q p and Op is unique because Op is algebraically closed, as wel l as

complete.

With the above definitions, we define the Newton polygon for

polynomials in p-adic field as given by Koblitz (1977) as fol lows:

4

Let [(x) = 1 + L�=l ai Xi be a polynomial of degree n with coefficients

in Or and constant tenn 1. Consider the points (i, or� �), if ai = 0, we omit that

point. The Newton polygon of f(x) is defined to be the "convex hull" of this set of

points which is constructed by taking a vertical line through (0, 0) and rotating

it about (0, 0) counterclockwise until it hits any of the points (i, or� aj ) and

finally hits the point (n, ordp �,).

Background

The role of the Newton polygon in obtaining properties of zero of

polynomials in one variable is quite well known. For example, the Newton

polygon can be usefully applied in proving Puiseux's theorem (Walker, 1962).

A. Sathaye (1983) also consider generalise Newton-Puiseux expansion.

Koblitz (1977) discusses the Newton polygon in the p-adic case for

polynomials and power series in np[x]. Here estimates concerning zeros of

polynomials are derived from the properties of the associated Newton polygon.

In particular, if A is the slope of a segment in the Newton polygon of a

polynomial f having length N, then there are N roots of f whose p-adic order is

-A.

5

For each prime p, let f = (f I, ... ,fn) be an n-tuple of polynomials in the

p-adic ring Zp [�] where � = (xt, ... ,xn). We consider the set

V(f; pU ) = {y mod pU : fey) == Q mod pU }

and denote N(f; pU) the cardinality of V(f; p<X) where a >0 and y runs through a

complete set of residues modulo pa .

Loxton and Smith (1982) investigate the application of Newton polygon

technique but finally the following method is used to arrive at their result.

With K as the algebraic number field generated by the roots �i , 1 � i � m

of the polynomial f(x) in Z[x], Loxton and Smith showed that

N(f; pU ) = m pU - (a-8)/e

if a> 8, where m is the number of distinct roots of f(x) and 8 = ordp D(f), where

D(i) denotes the intersections of the functional ideals of K generated by the

number

i> 1.

and e = max ej, with ej as the multiplicity of the roots �j .

By using a version of Hensel's Lemma, Chalk and Smith (1982)

obtain a result of similar form with 8 = maXi ordp fi where fi is the Taylor

coefficient

f(ei)(�i) e· , 1 •

at the distinct roots �i.

Loxton and Smith ( 1 982) show that for f= (fl' . . . '[.,)

if a � 28 if a> 28

6

where (5 = ordp D(i) and DCi) denotes the discriminant of f, and Deg f means the

product of the degrees of all the components of f.

Mohd. Atan and Loxton ( 1 986) extend the Newton polygon idea in the

p-adic case to polynomials in two variables and caIl it Newton polyhedron

method. Mohd. Atan (1986) investigates the relationships between roots of a

polynomial in Op [x,y] and its Newton polyhedron by considering the

combinations of the associated indicator diagrams.

He conjectures that to every simple point of intersection in the

combination of the indicator diagrams there exists common zero of both

polynomials whose p-adic order corresponds to this point. He then proves that if

7

(A,/l) is a point of intersection of the indicator diagrams associated with

polynomials f and g in Zp [x,y), which i s not a vertex of either diagram and

suppose that the edges through (A, /l) do not coincide, then there are S and 11 in

Let A be the matrix representing f the l inear polynomials with coefficients

in the p-adic ring Zp and a > 0, Mohd. Atan ( 1 988) shows that

if a�b if a>b

where 8 indicates the minimum of the p-adic orders of r X r non-singular

submatrices of A. He also shows that

if a�S if a>b

where f and g are l inear polynomials in Zp [x,y] with a > ° and 8 = or� J fg , the p-

adic order of the Jacobian of f and g.

8

Mohd. Atan ( 1 988) considers in particular, the non-linear polynomial

f= «( ,fy ) where f x' f yare the usual partial derivatives with respect to x and

y respectively of the polynomial

f(x,y) = ax3 + bx2y + cx + dy + e

in Zp [x,y] and give the estimate for N( f, ; fy ; p( l) explicitly In temlS of the

p-adic orders of the coefficients of f(x,y) as follows :

if a � 6 if a>6

where 0 = max {ordp 3a, 3/2 or� b} .

Mohd. Atan and Abdullah ( 1 992) consider a cubic polynomial of the

fonn

f(x,y) = ax3 + bx2 y + cxy2 + ely3 + kx + my + n

and obtained a result of similar form with 0 = max {ordp3a, or� b} . In both

cases, the method is first to reduce both polynomials fx , fy to polynomials in one

variable and next to consider combination of indicator diagrams associated with

the p-adic Newton polyhedrons of each polynomial to detennine the common

zeros of the polynomials .

9

Mohd. Atan and Abdullah ( 1 993) consider the same cubic polynomials

and obtain a result of similar fonn with 3 = {ordp3a, ordl' b, ordp c, ordp3d} .

They have found that the value of the determining factor 3 is in fact dependent

on the dominant terms of f. This gives a more symmetric result than the previous

one.

Organization of The Study

In this thesis we consider the set of solutions of congruence equations

modulo a prime power p associated with the polynomial

f(x ,y) = ax4 + bx3y + cx2 l + dxl + e/ + mx + ny + k

and its cardinality is then estimated by exammmg the Newton polygon

analogue for polynomials in Zp [x,y] .

We begin in the next chapter by discussing the polynomial rIngs,

i l lustrate the arithmetic operations of two polynomials and the degree for

polynomial in one variable as well as polynomial with multi-indetem1inates.

Then we examine the relationship between the roots of polynomial and the

derivatives of the same polynomial.