A note on mediated simplices

doi:10.1016/j.jpaa.2020.106608

Journal of Pure and Applied Algebra

Volume 225, Issue 7, July 2021, 106608

https://doi.org/10.1016/j.jpaa.2020.106608 Get rights and content

Abstract

Many homogeneous polynomials that arise in the study of sums of squares and Hilbert's 17th problem are those formed by monomial substitutions into the arithmetic-geometric inequality. In 1989, Reznick [14] gave a necessary and sufficient condition for such a form to have a representation as a sum of squares of forms, involving the arrangement of lattice points in the simplex whose vertices were the n-tuples of the exponents used in the substitution. Further, a claim was made, and not proven, that sufficiently large dilations of any such simplex will also satisfy this condition. The aim of this short note is to prove the claim, and provide further context for the result, both in the study of Hilbert's 17th Problem and the study of lattice point simplices.

Introduction

In 1989, the second author considered [14] a class of homogeneous polynomials (forms) which had arisen in the study of Hilbert's 17th Problem as monomial substitutions into the arithmetic-geometric inequality. The goal was to determine when such a form, which must be positive semidefinite, had a representation as a sum of squares of forms. The answer was a necessary and sufficient condition involving the arrangement of lattice points in the simplex whose vertices were the n-tuples of the exponents used in the substitution. Further, a claim was made in [14], and not proven, that sufficiently large dilations of any such simplex will also satisfy this condition. The aim of this short note is to prove the claim, and provide further context for the result, both in the study of Hilbert's 17th Problem and the study of lattice point simplices. The second author is happy to acknowledge that the return to this claim was triggered by two nearly simultaneous events: an invitation to speak at the 2019 SIAM Conference on Applied Algebraic Geometry, and a request from Jie Wang for a copy of [15], which was announced in [14] but never written. The second author was supported in part by Simons Collaboration Grant 280987.

Section snippets

Preliminaries

We work with homogeneous polynomials (forms) in $R [x] = R [x_{1}, \dots, x_{n}]$ , the ring of real polynomials in n variables. Write the monomial $x_{1}^{α_{1}} \dots x_{n}^{α_{n}}$ as $x^{α}$ , for $α = (α_{1}, \dots, α_{n}) \in Z^{n}$ . For $S \subset Z^{n}$ , $c v x (S)$ denotes the convex hull of S. For $p (x) = \sum_{α} c (α) x^{α} \in R [x]$ , let supp $(p) = {α | c (α) \neq 0}$ , write $New (p)$ for the Newton polytope of p, that is, $New (p) = c v x ($ supp $(p))$ , and let $C (p) = New (p) \cap Z^{n}$ .

A form $p \in R [x]$ is positive semidefinite or psd if $p (x) \geq 0$ for all $x \in R^{n}$ . It is a sum of squares or sos if $p = \sum_{j} h_{j}^{2}$ for forms $h_{j} \in R [x]$ . Clearly,

Main theorem

The following theorem was asserted in [14, Prop. 2.7].

Theorem 3.1

For every integer $k \geq \max {2, m - 2}$ , $c v x (k U) \cap Z^{n}$ is $(k U)$ -mediated.

Corollary 3.2

Any agiform $p \in R [x_{1}, \dots, x_{n}]$ can be written as a sum of squares of forms in the variables $x_{i}^{1 / k}$ for $k \geq \max {2, m - 2}$ .

To prove Theorem 3.1, we show that any non-vertex $w \in c v x (k U) \cap Z^{n}$ is the average of two different points in $c v x (k U) \cap {(2 Z)}^{n}$ . The proof for $k \geq m - 1$ is easy, while the proof for the remaining case ( $m \geq 4$ and $k = m - 2$ ) is more delicate. We defer the discussion of Corollary 3.2 to the next

Implication for Hilbert's 17th problem

Proof of Corollary 3.2

Suppose $p (x) = λ_{1} x^{u_{1}} + \dots + λ_{n} x^{u_{n}} - x^{w}$ . Let $q (x_{1}, \dots, x_{n}) : = p (x_{1}^{k}, \dots, x_{n}^{k}) = λ_{1} x^{k u_{1}} + \dots + λ_{n} x^{k u_{n}} - x^{k w},$ which is also an agiform. By Theorem 2.1, Theorem 3.1, q is sos, and so $q = \sum_{j = 1}^{r} h_{j}^{2} \Rightarrow p (x_{1}, \dots, x_{n}) = \sum_{j = 1}^{r} h_{j}^{2} (x_{1}^{1 / k}, \dots, x_{n}^{1 / k}),$ which shows that p has the desired representation. □

At the time that [14] was written, and the proof given here was relegated to the proposed preprint [15], the second author entertained the possibility that such a result might be true for any psd form. Unfortunately, he discovered that the so-called

Implication for polytopes

From the point of view of polytopes, one would more naturally write $U = 2 P$ , where P is a lattice-point simplex in $R^{n}$ ; without loss of generality, we also assume $m = n$ . Further, the conditions that the vertices lie on a hyperplane and have non-negative coefficients seem artificial. In this way, we can drop n-th component, so that $P$ is the usual n-point lattice simplex in $R^{n - 1}$ .

Let $d = n - 1$ . Then Theorem 3.1 says that if $k \geq \max {2, d - 1}$ , then a non-vertex $w \in 2 k P \cap Z^{d}$ can be written as a sum of two different

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Journal of Pure and Applied Algebra

A note on mediated simplices

Abstract

Introduction

Section snippets

Preliminaries

Main theorem

Implication for Hilbert's 17th problem

Implication for polytopes

On non-negative forms in real variables some or all of which are non-negative

Proc. Camb. Philos. Soc.

A Positivstellensatz for sums of nonnegative circuit polynomials

SIAM J. Appl. Algebra Geom.

The algebraic boundary of the sonc cone

Copositive and completely positive quadratic forms

Proc. Camb. Philos. Soc.

Integral body-building in R3

J. Geom.

Positive Polynomials, Convex Integral Polytopes and a Random Walk Problem

Über die Darstellung definiter Formen als Summe von Formenquadraten

Math. Ann.

Integral body-building in $R^{3}$