Moments, Sums of Squares, and Tropicalization

TL;DR

This paper employs tropicalization to analyze the duals of cones of nonnegative polynomials and sums of squares, providing combinatorial descriptions and revealing limitations of sums of squares approximations.

Contribution

It introduces explicit tropical descriptions of moment and pseudo-moment cones and explores their properties, especially for binomial-defined semialgebraic sets.

Findings

Tropicalizations distinguish nonnegative polynomials from sums of squares.

Explicit combinatorial descriptions of tropicalized cones are provided.

Identifies limitations of sums of squares approximations.

Abstract

We use tropicalization to study the duals to cones of nonnegative polynomials and sums of squares on a semialgebraic set $S$. The truncated cones of moments of measures supported on the set $S$ is dual to nonnegative polynomials on $S$, while "pseudo-moments" are dual to sums of squares approximations to nonnegative polynomials. We provide explicit combinatorial descriptions of tropicalizations of the moment and pseudo-moment cones, and demonstrate their usefulness in distinguishing between nonnegative polynomials and sums of squares. We give examples that show new limitations of sums of squares approximations of nonnegative polynomials. When the semialgebraic set is defined by binomial inequalites, its moment and pseudo-moment cones are closed under Hadamard product. In this case, their tropicalizations are polyhedral cones that encode all binomial inequalities on the moment and pseudo-moment cones.