Symmetric Sums of Squares over $k$-Subset Hypercubes

TL;DR

This paper develops a symmetry-reduction method for sum of squares representations of symmetric polynomials over k-subset hypercubes, connecting algebraic techniques with combinatorial flag algebra concepts.

Contribution

It introduces a simplified, flexible approach to sum of squares proofs that depends only on degree, not the number of variables, and links algebraic certificates with flag algebra theory.

Findings

Every symmetric polynomial with a fixed-degree sos has a succinct sos expression independent of variable count.

The method provides a representation-theoretic justification for Razborov's flags.

Identifies non-negative polynomials that cannot be certified with any fixed set of flags.

Abstract

We consider the problem of finding sum of squares (sos) expressions to establish the non-negativity of a symmetric polynomial over a discrete hypercube whose coordinates are indexed by the $k$-element subsets of $[n]$. For simplicity, we focus on the case $k=2$, but our results extend naturally to all values of $k \geq 2$. We develop a variant of the Gatermann-Parrilo symmetry-reduction method tailored to our setting that allows for several simplifications and a connection to flag algebras. We show that every symmetric polynomial that has a sos expression of a fixed degree also has a succinct sos expression whose size depends only on the degree and not on the number of variables. Our method bypasses much of the technical difficulties needed to apply the Gatermann-Parrilo method, and offers flexibility in obtaining succinct sos expressions that are combinatorially meaningful. As a byproduct of our results, we arrive at a natural representation-theoretic justification for the concept of flags as introduced by Razborov in his flag algebra calculus. Furthermore, this connection exposes a family of non-negative polynomials that cannot be certified with any fixed set of flags, answering a question of Razborov in the context of our finite setting.