A note on mediated simplices

TL;DR

This paper proves a claim about the behavior of dilated simplices related to sums of squares representations of polynomials, providing insights into Hilbert's 17th problem and lattice point geometry.

Contribution

It proves that sufficiently large dilations of certain simplices satisfy a key condition for sums of squares representations, confirming a previously unproven claim.

Findings

Large dilations of simplices satisfy the sum of squares condition

Provides new context for Hilbert's 17th problem

Enhances understanding of lattice point arrangements in simplices

Abstract

Many homogeneous polynomials that arise in the study of sums of squares and Hilbert's 17th problem come from monomial substitutions into the arithmetic-geometric inequality. In 1989, the second author gave a necessary and sufficient condition for such a form to have a representation as a sum of squares of forms (Math. Ann., (283), 431--464), 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.