Aces Polya Talk 2

8/4/2019 Aces Polya Talk 2

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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
http://find/http://goback/


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Plyas Theorem

Idea of Plyas Proof





Outline

1 Plyas Theorem



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Plyas Theorem

Idea of Plyas Proof





Outline

1 Plyas Theorem

2 Idea of Plyas Proof



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Plyas Theorem

Idea of Plyas Proof





Outline

1 Plyas Theorem


3 History and Significance



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Plyas Theorem

Idea of Plyas Proof





Outline

1 Plyas Theorem



4 Applications of Plyas Theorem


Pl Th


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Plyas Theorem

Idea of Plyas Proof





Outline

1 Plyas Theorem




5 Degree Bounds for Plyas Theorem

For strictly positive polynomialsFor polynomials with corner zeros


Pl Th


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Plyas Theorem

Idea of Plyas Proof





Outline

1 Plyas Theorem






6 A Characterization of Polynomials Satisfying Plyas

Theorem


Plyas Theorem


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Plya s Theorem

Idea of Plyas Proof





Outline

1 Plyas Theorem







Theorem


Plyas Theorem


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Plya s Theorem

Idea of Plyas Proof





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.


Plyas Theorem


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Plya s Theorem

Idea of Plyas Proof





Outline

1 Plyas Theorem







Theorem


Plyas Theorem


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Plya s Theorem

Idea of Plyas Proof





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.


Plyas Theorem


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Plya s Theorem

Idea of Plyas Proof





Outline

1 Plyas Theorem







Theorem


Plyas Theorem


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Idea of Plyas Proof





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.


Plyas Theorem


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Idea of Plyas Proof





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.


Plyas Theorem


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Idea of Plyas Proof





Outline

1 Plyas Theorem







Theorem


Plyas Theorem


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Idea of Plyas Proof





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.


Plyas Theorem

Id f Pl P f


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Idea of Plyas Proof





For strictly positive polynomials

For polynomials with corner zeros

Outline

1 Plyas Theorem







Theorem


Plyas Theorem

Id f Pl P f


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Idea of Plyas Proof







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.


Plyas Theorem

Idea of Plyas Proof


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Idea of Plya s Proof







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}


Plyas Theorem

Idea of Plyas Proof


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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.


Plyas TheoremIdea of Plyas Proof


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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()




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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.




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Outline

1 Plyas Theorem








Theorem




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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.




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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.




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