Dimensional Differences Between Faces of the Cones of Nonnegative Polynomials and Sums of Squares

TL;DR

This paper investigates the dimensional differences between faces of the cones of nonnegative polynomials and sums of squares, revealing that these gaps are common and characterizing when they occur.

Contribution

It provides a complete characterization of when dimensional gaps occur between these cones for ternary forms and quaternary quartics, including explicit descriptions in key cases.

Findings

Dimensional gaps occur in all cases with nonnegative polynomials not expressible as sums of squares.

These gaps are generic and not due to special face selection.

Explicit descriptions are provided for the smallest cases where the cones differ.

Abstract

We study dimensions of the faces of the cone of nonnegative polynomials and the cone of sums of squares; we show that there are dimensional differences between corresponding faces of these cones. These dimensional gaps occur in all cases where there exist nonnegative polynomials that are not sums of squares. The gaps occur generically, they are not the product of selecting special faces of the cones. For ternary forms and quaternary quartics, we completely characterize when these differences are observed. Moreover, we provide an explicit description for these differences in the two smallest cases, in which the cone of nonnegative polynomials and the cone of sums of squares are different. Our results follow from more general results concerning the relationship between the second ordinary power and the second symbolic power of the vanishing ideal of points in projective space.