There are Significantly More Nonnegative Polynomials than Sums of Squares

TL;DR

This paper demonstrates that as the number of variables increases, the set of nonnegative polynomials vastly outnumbers sums of squares, with quantitative bounds showing the ratio diminishes to zero.

Contribution

It provides the first quantitative comparison of the sizes of the cones of nonnegative polynomials and sums of squares in high dimensions.

Findings

The volume ratio of the cones tends to zero as variables increase.

Nonnegative polynomials are significantly more abundant than sums of squares in high dimensions.

The results hold for fixed degree greater than 2.

Abstract

We investigate the quantitative relationship between nonnegative polynomials and sums of squares of polynomials. We show that if the degree is fixed and the number of variables grows then there are significantly more nonnegative polynomials than sums of squares. More specifically, we take compact bases of the cone of nonnegative polynomials and the cone of sums of squares and derive bounds for the volumes of the bases. If the degree is greater than 2 then we show that the ratio of the volumes of the bases, raised to the power reciprocal to the ambient dimension, tends to 0 as the number of variables tends to infinity.