Sums of squares and varieties of minimal degree

TL;DR

This paper characterizes when nonnegative quadratic forms on certain projective varieties are sums of squares, extending Hilbert's classical result, and classifies related lattice polytopes for sums of squares representations.

Contribution

It provides a geometric criterion linking varieties of minimal degree to sums of squares representations of nonnegative forms, extending classical results.

Findings

Nonnegative quadratic forms on varieties of minimal degree are sums of squares.

Complete classification of cases where nonnegative Laurent polynomials are sums of squares.

Identification of lattice polytopes where all nonnegative Laurent polynomials are sums of squares.

Abstract

Let X be a real nondegenerate projective subvariety such that its set of real points is Zariski dense. We prove that every real quadratic form that is nonnegative on X is a sum of squares of linear forms if and only if X is a variety of minimal degree. This substantially extends Hilbert's celebrated characterization of equality between nonnegative forms and sums of squares. We obtain a complete list for the cases of equality and also a classification of the lattice polytopes Q for which every nonnegative Laurent polynomial with support contained in 2Q is a sum of squares.