Volumes of Nonnegative Polynomials, Sums of Squares and Powers of Linear Forms

TL;DR

This paper investigates the volume relationships among cones of nonnegative polynomials, sums of squares, and sums of powers of linear forms, revealing asymptotic discrepancies as the number of variables grows.

Contribution

It provides asymptotically exact bounds on the volumes of sections of these cones and quantifies the growing discrepancy between them with increasing variables.

Findings

Nonnegative polynomials vastly outnumber sums of squares for degree > 2.

Sums of squares significantly outnumber sums of powers of linear forms.

Discrepancy between cones increases with the number of variables.

Abstract

We study the quantitative relationship between the cones of nonnegative polynomials, cones of sums of squares and cones of sums of powers of linear forms. We derive bounds on the volumes (raised to the power reciprocal to the ambient dimension) of compact sections of the three cones. We show that the bounds are asymptotically exact if the degree is fixed and number of variables tends to infinity. When the degree is larger than two it follows that there are significantly more non-negative polynomials than sums of squares and there are significantly more sums of squares than sums of powers of linear forms. Moreover, we quantify the exact discrepancy between the cones; from our bounds it follows that the discrepancy grows as the number of variables increases.