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Plyas Theorem
Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Everything youve ever wanted to know about
Plyas Theorem
(but were afraid to ask).
Mari Castle
Emory University, Atlanta
December 2, 2006
M. Castle Plyas Theorem
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Plyas Theorem
Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Outline
1 Plyas Theorem
M. Castle Plyas Theorem
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Plyas Theorem
Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Outline
1 Plyas Theorem
2 Idea of Plyas Proof
M. Castle Plyas Theorem
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Plyas Theorem
Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Outline
1 Plyas Theorem
2 Idea of Plyas Proof
3 History and Significance
M. Castle Plyas Theorem
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Plyas Theorem
Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Outline
1 Plyas Theorem
2 Idea of Plyas Proof
3 History and Significance
4 Applications of Plyas Theorem
M. Castle Plyas Theorem
Pl Th
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Plyas Theorem
Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Outline
1 Plyas Theorem
2 Idea of Plyas Proof
3 History and Significance
4 Applications of Plyas Theorem
5 Degree Bounds for Plyas Theorem
For strictly positive polynomialsFor polynomials with corner zeros
M. Castle Plyas Theorem
Pl Th
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Plyas Theorem
Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Outline
1 Plyas Theorem
2 Idea of Plyas Proof
3 History and Significance
4 Applications of Plyas Theorem
5 Degree Bounds for Plyas Theorem
For strictly positive polynomialsFor polynomials with corner zeros
6 A Characterization of Polynomials Satisfying Plyas
Theorem
M. Castle Plyas Theorem
Plyas Theorem
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Plya s Theorem
Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Outline
1 Plyas Theorem
2 Idea of Plyas Proof
3 History and Significance
4 Applications of Plyas Theorem
5 Degree Bounds for Plyas Theorem
For strictly positive polynomialsFor polynomials with corner zeros
6 A Characterization of Polynomials Satisfying Plyas
Theorem
M. Castle Plyas Theorem
Plyas Theorem
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Plya s Theorem
Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Plyas Theorem
Fix a positive integer n and let R[X] = R[x1, . . . , xn].
Let n denote the simplex {(x1, . . . , xn)|xi 0,
i xi = 1}.
Let p R[X] be a homogeneous polynomial.
Theorem (Plya)
If p is positive onn, then for sufficiently large N N all thecoefficients of(x1 + ... + xn)
Np(x1, ...,xn) are positive.
M. Castle Plyas Theorem
Plyas Theorem
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Plya s Theorem
Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Outline
1 Plyas Theorem
2 Idea of Plyas Proof
3 History and Significance
4 Applications of Plyas Theorem
5 Degree Bounds for Plyas Theorem
For strictly positive polynomialsFor polynomials with corner zeros
6 A Characterization of Polynomials Satisfying Plyas
Theorem
M. Castle Plyas Theorem
Plyas Theorem
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Plya s Theorem
Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
A brief sketch of Plyas proof
Let p R[X] be homogeneous and positive on n withdeg(p) = d. Plya constructs a sequence of real polynomials
p such that
1 p converges uniformly to p on n as 0
2 for || = N + d, the coefficient of X in (xi)N p is a
non-negative multiple of p 1N+d
( N+d)
Since N+d n, for large enough N the coefficient of X is
non-negative.
M. Castle Plyas Theorem
Plyas Theorem
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Plya s Theorem
Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Outline
1 Plyas Theorem
2 Idea of Plyas Proof
3 History and Significance
4 Applications of Plyas Theorem
5 Degree Bounds for Plyas Theorem
For strictly positive polynomialsFor polynomials with corner zeros
6 A Characterization of Polynomials Satisfying Plyas
Theorem
M. Castle Plyas Theorem
Plyas Theorem
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y
Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
History
In 1928, Plyas Theorem was published in German and also in
English in Inequalitiesby Hardy, Littlewood, and Plya. They
write
The theorem gives a systematic process for deciding
whether a given form F is strictly positive for positive
x. We multiply repeatedly by
xi and, if the form is
positive, we shall sooner or later obtain a form withpositive coefficients.
M. Castle Plyas Theorem
Plyas Theorem
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Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Significance
Plyas Theorem implies that for p R[X] homogeneous and
positive on n, p can be written as a quotient of twopolynomials with positive coefficients since p = f(x1+...+xn)N
,
where f has positive coefficients.
If we replace the xi with x2i (so n becomes the unit sphere) we
have a constructive solution to Hilberts 17th problem,whenever p is even and definite.
M. Castle Plyas Theorem
Plyas Theorem
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Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Outline
1 Plyas Theorem
2 Idea of Plyas Proof
3 History and Significance
4 Applications of Plyas Theorem
5 Degree Bounds for Plyas Theorem
For strictly positive polynomialsFor polynomials with corner zeros
6 A Characterization of Polynomials Satisfying Plyas
Theorem
M. Castle Plyas Theorem
Plyas Theorem
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Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Applications
Plyas Theorem has been used to
prove directly that a positive definite form is a sum of
squares of rational functions,
give an algorithmic proof of Schmdgens
Positivstellensatz,
prove that the moment problem is solvable in certain cases,
find optimization solutions in copositive programming,
approximate the stability number of a graph.
M. Castle Plyas Theorem
Plyas Theorem
Id f Pl P f
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Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
For strictly positive polynomials
For polynomials with corner zeros
Outline
1 Plyas Theorem
2 Idea of Plyas Proof
3 History and Significance
4 Applications of Plyas Theorem
5 Degree Bounds for Plyas Theorem
For strictly positive polynomialsFor polynomials with corner zeros
6 A Characterization of Polynomials Satisfying Plyas
Theorem
M. Castle Plyas Theorem
Plyas Theorem
Id f Pl P f
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Idea of Plyas Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
For strictly positive polynomials
For polynomials with corner zeros
Degree BoundIn 2001, V. Powers and B. Reznick gave a bound for N.
Suppose p R[X] is homogeneous and positive on n withdeg(p) = d. Assume p = ||=daX
and define the following
:= min{p(X) : X n}L := max{|a|
1!...n!||! }
Theorem (V. Powers, B. Reznick)
Suppose p is as above. If
N>d(d 1)
2
L
d
then(x1 + ... + xn)Np(x1, ..., xn) has positive coefficients.
M. Castle Plyas Theorem
Plyas Theorem
Idea of Plyas Proof
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Idea of Plya s Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
For strictly positive polynomials
For polynomials with corner zeros
Degree Bound
In 2006, V. Powers and B. Reznick extended their results to
non-negative polynomials allowed to have zeros at the corners
of n.
Definition
Let p(X) R[X] be a form of degree d and supposep(X) 0 on n. Then p has a simple zeroat the unitvector ej if the coefficient of x
dj in p is zero, but the
coefficient of xd1j xi is non-zero (and necessarily positive)for each i= j.
For r R,0 < r < 1 and j = 1, ....,n, let n(j, r) denote thesimplex with vertices {ej} {ej + r((ei ej)|i= j}
M. Castle Plyas Theorem
Plyas Theorem
Idea of Plyas Proof
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Idea of Plya s Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
For strictly positive polynomials
For polynomials with corner zeros
Degree Bound
Lemma
Suppose p R[X] is a form of degree d with p(X) 0 onn. Ifp has a simple zero at ej, then there exist s, r > 0 such that
p(u1, ...,un) s(u1 + ... + uj1 + uj+1 + ... + un)
for all u = (u1, ...,un) n(j, r). More precisely, let C be thesum of the absolute value of the coefficients of p and let
v = mini=j pxi (ej), then we can take
r =v
v + 2C, s =
v
2(
2C
v + 2C)d1.
M. Castle Plyas Theorem
Plyas TheoremIdea of Plyas Proof
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Idea of Plya s Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
For strictly positive polynomials
For polynomials with corner zeros
Degree Bound
Suppose p R[X] is homogeneous with deg(p) = d andpositive on n except for a simple zero at e1. Let s, r > 0 be asin the lemma. Assume p = ||=daX
and define the following
K is the closure of n with the corner n(1, r) removed
:= min{p(X) : X K}
L := max{|a|1!...n!||! }
M := max{|a| : a = 0}
For = (1,...,n) Nn
c() :=
i2i(i1)
2
c :=
a=0c()
M. Castle Plyas Theorem
Plyas TheoremIdea of Plyas Proof
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Idea of Plya s Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
For strictly positive polynomials
For polynomials with corner zeros
Degree Bound
Proposition (V. Powers, B. Reznick)
Suppose p is positive on n except for a simple zero at e1. If
N> max{d(d 1)
2
L
d,
cM
s d},
then the coefficient of X in(x1 + ... + xn)Np is nonnegative.
M. Castle Plyas Theorem
Plyas TheoremIdea of Plyas Proof
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Idea of Plya s Proof
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Outline
1 Plyas Theorem
2 Idea of Plyas Proof
3 History and Significance
4 Applications of Plyas Theorem
5 Degree Bounds for Plyas Theorem
For strictly positive polynomials
For polynomials with corner zeros
6 A Characterization of Polynomials Satisfying Plyas
Theorem
M. Castle Plyas Theorem
Plyas TheoremIdea of Plyas Proof
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y
History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
A recent characterization
In 2005, M. Schweighofer applied Plyas Theorem to several
new environments including positive semidefinite polynomials,
not necessarily homogeneous, with a certain representation.
Lemma
Let f R[X]. Suppose that for every x n there are m N,forms g1, ...,gm and h1, ...,hm R[X]
+ such that
1) f = g1h1 + ... + gmhm2) gi > 0 for all x nThen there exists N N such that(x1 + ... + xn)
Nf has
nonnegative coefficients.
M. Castle Plyas Theorem
Plyas TheoremIdea of Plyas Proof
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History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
Plya Quotes
What is the difference between method and device? A
method is a device which you used twice.
If there is a problem you cant solve, then there is an easier
problem you can solve: find it.
How I need a drink, alcoholic of course, after the heavy
chapters involving quantum mechanics.
M. Castle Plyas Theorem
Plyas TheoremIdea of Plyas Proof
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History and Significance
Applications of Plyas Theorem
Degree Bounds for Plyas Theorem
A Characterization of Polynomials Satisfying Plyas Theorem
QUESTIONS?
M. Castle Plyas Theorem
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