Sums of Squares on the Hypercube

TL;DR

This paper investigates sum of squares representations of nonnegative polynomials on finite sets, providing bounds and constructions that highlight the complexity of such representations, especially on the hypercube, with implications for polynomial optimization.

Contribution

It derives a tight upper bound for sum of squares rational representations on the hypercube and constructs polynomials requiring increasing degrees, advancing understanding of Hilbert's 17th problem.

Findings

Upper bound for rational sum of squares representations based on Hilbert function

Construction of non-sos quartic polynomials on the hypercube requiring higher degrees

First known family of bounded degree polynomials needing increasing rational function degrees

Abstract

Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we are interested in rational function representations of small degree. We derive a general upper bound in terms of the Hilbert function of X, and we show that this upper bound is tight for the case of quadratic functions on the hypercube C={0,1}^n, a very well studied case in combinatorial optimization. Using the lower bounds for C we construct a family of globally nonnegative quartic polynomials, which are not sums of squares of rational functions of small degree. To our knowledge this is the first construction for Hilbert's 17th problem of a family of polynomials of bounded degree which need increasing degrees in rational function representations as the number of variables n goes to infinity. We note that representation theory of the symmetric group S_n play a crucial role in our proofs of the lower bounds.