Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Polya’s Positivity Proposition

Vicki Powers

Dept. of Mathematics and Computer ScienceEmory University, Atlanta, USA

September 8, 2005, Fourth annual network meeting RAAG2005

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

1 Polya’s Theorem

2 Some history

3 Polya’s Theorem with degree bounds

4 Applications

5 Generalizations

6 Polya’s Theorem with zeros

7 Polya Quotes

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Fix a positive integer n and let R[X ] := R[x1, . . . , xn]. We write∆n for the simplex {(x1, . . . , xn) | xi ≥ 0,

∑i xi = 1}.

Polya’s Theorem says that if a form (homogeneous polynomial)p ∈ R[X ] is positive on ∆n, then for sufficiently large N all thecoefficients of

(x1 + · · ·+ xn)N ∗ p(x1, . . . , xn)

are positive.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

In 1883, Poincare proved for n = 2 that for a form p positiveon ∆2 there is some form q with positive coefficients so thatq ∗ p has only positive coefficients.

In 1911, Meissner proved it for n = 3 but again withoutspecifying the q.

Polya’s theorem appeared in 1928 (in German) and is also inInequalities by Hardy, Littlewood, and Polya (in English),where they write

The theorem gives a systematic process for decidingwhether a given form F is strictly positive forpositive x. We multiply repeatedly by

∑xi and, if

the form is positive, we shall sooner or later obtain aform with positive coefficients.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Polya’s proof is elementary: For p positive on ∆n of degree d , heconstructs a sequence of real polynomials pε in n variables suchthat

pε converges uniformly to p on ∆n as ε → 0 and

for |β| = N + d , the coefficient of X β in (∑

xi )N ∗ p is a

non-negative multiple of p 1N+d

( βN+d )

Since βN+d ∈ ∆n, for large enough N the coefficient of X β is

non-negative.

Remark. An alternate proof of Polya’s Theorem was given byT. Wormann (circa 2000) using the Representation Theorem formodules over archimedean preprimes.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Polya’s Theorem says that any form p positive on ∆n can bewritten as a quotient of two polynomials with positive coefficientswhere the denominator is a power of

∑i xi .

If we replace xi by x2i , ∆n becomes the unit sphere and the

theorem gives a concrete, constructive solution to Hilbert’s 17thproblem in the case that p is even and definite. In his 1928 paper,Polya remarks

Es kann schliesslich bemerkt werden, dass die Darstellungeinigermassen in Zusammenhang mit einer Fragestellungvon Hilbert steht, die kurzlich durch E. Artin mittiegehenden Mittleln gelost wurde.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

In 2001, a bound for Polya’s Theorem was given. Supposep ∈ R[X ] is a form of degree d positive on ∆n, let λ be theminimum of p on ∆n. Assume p =

∑|α|=d aαXα and define

L := maxα

|aα|α1! . . . αn!

|α|!

Theorem (V. Powers,B. Reznick)

Suppose that p ∈ R[X ] is as above. If

N >d(d − 1)

2

L

λ− d ,

then (x1 + · · ·+ xn)N ∗ p(x1, . . . , xn) has positive coefficients.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Applications

In 1940, Habicht use Polya’s Theorem to prove directly that apositive definite form is a sum of squares of rational functions.

In 2002, M. Schweighofer used Polya’s Theorem to give aalgorithmic proof of Schmudgen’s Positivstellensatz: If the basicclosed semialgebraic set K = {g1 ≥ 0, . . . , gk ≥ 0} is compact andf > 0 on K , then f is in the preorder generated by the gi ’s.By the (classic) Positivstellensatz, there is an explicit “certificateof compactness” for K . Modulo such a certificate, for f > 0 on KSchweighofer gives a method for constructing a representation of fin the preorder by reducing to Polya’s Theorem.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Combining this construction with the degree bound for Polya’sTheorem yields a complexity estimate for Schmudgen’sPositivstellensatz.

This construction can be generalized to cylinders with compactcross-section – sets of the form K × L, where K is a compact basicclosed semialgebraic set and L ⊆ R is non-compact – and used toprove that the moment problem is solvable in this case. (Themoment problem in this case was also solved by Kuhlmann,Marshall, and Schwartz using different methods.)

Very recently, J. Nie generalized this construction to give analgorithmic proof of Putinar’s Theorem with complexity bound.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Application to copositive programming: Let Sn denote the n × nsymmetric matrices over R and define the copositive cone

Cn = {M ∈ Sn | Y TMY ≥ 0 for all Y ∈ Rn+}.

Copositive programming, i.e., optimization over Cn, has manyapplications. For example, if G is a loopless undirected graph withvertex set V let α(G ) denote the stability number of G (the size ofthe maximum stable subset of V ). Then

Theorem (De Klerk, Pasechnik)

Let n = |V | and A(G ) be the adjacency matrix of G. Thenα(G ) = min{λ | λ(I + A(G ))− 1 ∈ Cn}.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Now define, for r ∈ N,

C rn = {M ∈ Sn | (

∑i

xi )r ∗ XTMX has non-negative coefficients }.

Then C rn can be computed via linear programming and by

Polya’s Theorem C rn converges to Cn.

De Klerk and Pasechnik use the bound for Polya’s Theorem togive results on using this method for approximating thestability number of a graph.

Note that this is in the same spirit as using SOSrepresentations of positive polynomials to minimizepolynomials on semialgebraic sets via semidefiniteprogramming.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Generalizations

In 1969, Motzkin and Strauss partially generalized Polya’sTheorem to power sequences in several variables.

In 1996, Catlin and D’Angelo generalized the theorem topolynomials in several complex variables.

Handelman has results on a related question: Supposeq ∈ R[X ] has nonnegative coefficients, for which f does thereexist r ∈ N so that qr ∗ f has nonnegative coefficients?

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Recently, C.W.J. Hol and C.W. Scherer proved a generalization ofPolya’s Theorem (with bound) to a non-commutative setting.

The setting is m×m matrices with entries from R[X ], equivalently,polynomials in commuting variables with matrix coefficients.

Such S(X ) is sos if there is some T (X ) (not necessarily square)such that

S(X ) = T (X )TT (X ).

Note that if m = 1 and Tj(X ) is the j-th column of T , then thisbecomes S(X ) =

∑Tj(X )2 and we have the usual definition of

sos.

In this setting, the semialgebraic set is replaced by

{x ∈ Rn | −G (x) is psd },where G (X ) is a symmetric matrix with entries from R[X ]. Notethat if G (X ) = diag(−g1(X ), . . . ,−gm(X )), we obtain a basicclosed semialgebraic set.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Hol and Scherer generalize Polya’s Theorem with a bound in thissetting; the proof is similar to the commutative case.

They then give a matrix version of Putinar’s theorem, using ageneralization of Schweighofer’s procedure for reducing topositivity on the simplex.

Their result has applications to semi-definite programmingrelaxations of certain types of problems arising in control theory.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Recently, Schweighofer proved the following:

Lemma

Given f ∈ R[X ] and suppose that for each u ∈ ∆n there is m ∈ N,g1, . . . , gm, h1, . . . , hm ∈ R[X ] such that(1) f = g1h1 + · · ·+ gmhm

(2) gi (u) > 0 for all i(3) hi has non-negative coefficients for each iThen for some N, (

∑i xi )

N ∗ f has non-negative coefficients.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

In ongoing work with B. Reznick, we are looking at Polya’sTheorem for p psd on ∆n with possible zeros. We write Pn,d forthe forms in R[X ] of degree d for which the conclusion of Polya’sTheorem holds.

An easy argument yields the following:

Proposition

If p ∈ Pn,d , then p is positive on ∆n except possibly on a union offaces of ∆n.

For example, if p vanishes on the interior of ∆n, then p isindentically zero and if p(u) = 0 where u = (u1, . . . , uk , 0, . . . , 0)with ui > 0 for i = 1, . . . , k, then p vanishes on the facecontaining u.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Our future result: Suppose p ∈ Pn,d is zero on a face F of ∆n, wesay p has a simple zero on F if 〈a condition involving the positivityon F of certain partial derivatives of p〉 holds. Then the thetheorem we hope to prove is:

Suppose p is positive on ∆n except for possible simple zeros onfaces of ∆n. Then for N > 〈an explicit bound〉,

(x1 + · · ·+ xn)N ∗ p

has non-negative coefficients.

The theorem without the bound will most likely also followfrom Handelman’s work and Schweighofer’s lemma

The “simple zeros” condition is probably necessary as well assufficient; this should follow from Handelman’s work (or canbe proven directly)

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Let’s look at the case where p has a zero at a vertex of ∆n. Weassume throughout that the form p ∈ R[X ] is ≥ 0 on ∆n.

Definition

If p has a zero at a unit vector ej , it is a simple zero if for all i ,

i 6= j , ∂p∂xi

(ej) > 0, i.e., the coefficient of xd−1j xi is positive.

For r ∈ R, 0 < r < 1 and j = 1, . . . , n, let ∆n(j , r) denote thesimplex with vertices {ej} ∪ {ej + r(ei − ej) | i 6= j}.

Lemma

If p has a simple zero at ej , then there exists s > 0 such that

p(u1, . . . , un) ≥ s(u1 + · · ·+ uj−1 + uj+1 + · · ·+ un)

for all u = (u1, . . . , un) ∈ ∆n(j , r) for r sufficiently small.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Assume p has a simple zero at e1 and is otherwise positive on ∆n.

Choose s, r as in the lemma and let P be the closed simplexobtained by removing the ∆n(1, r) from ∆n and adding back theedges with vertices {e1, r(ei − e1)}.

By assumption, p > 0 on P. Let γ be the minimum of p on P.Recall from Polya’s proof that we need to find N such that

p 1(N+d)

(β

N + d

)> 0.

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

If βN+d ∈ P, then we can use the original Polya’s Theorem bound:

We need

N >d(d − 1)

2

L

γ− d .

If βN+d ∈ ∆n(1, r), then the lemma above combined with an

elementary argument shows that for s as in the lemma,T = max |aα| and a constant k which depends only on d , we need

N >kT

s− d .

Combining the two estimates gives a global bound on N.

Remark The bound s is not sensitive to the value of r , we justneed to make r small enough that the value of p on the corner isclose to the value of the derivatives. But the smaller r is, thesmaller γ will be, in general. The relationship between the twoestimates needs to be made precise!

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition

Outline Polya’s Theorem Some history Polya’s Theorem with degree bounds Applications Generalizations Polya’s Theorem with zeros Polya Quotes

Polya Quotes

If you can’t solve a problem, then there is an easier problem youcan solve: find it.

I am too good for philosophy and not good enough for physics.Mathematics is in between.

How I need a drink, alcoholic of course, after the heavy chaptersinvolving quantum mechanics(This is a mnemonic for the first fourteen digits of π, the lengthsof the words are the digits!)

Vicki Powers Dept. of Mathematics and Computer Science Emory University, Atlanta, USA

Polya’s Positivity Proposition