Low-Rank Sum-of-Squares Representations on Varieties of Minimal Degree

TL;DR

This paper generalizes Hilbert's result by showing that nonnegative quadratic forms on varieties of minimal degree can be expressed as sums of a minimal number of squares, with implications for low-rank factorizations and sum-of-squares representations.

Contribution

It establishes the minimal number of squares needed for nonnegative quadratic forms on varieties of minimal degree, extending previous results and providing a count of equivalence classes.

Findings

Every nonnegative quadratic form on a variety of minimal degree is a sum of im(X)+1$ squares.

The bound is proven to be optimal.

Results imply low-rank factorizations of positive semidefinite bivariate matrix polynomials.

Abstract

A celebrated result by Hilbert says that every real nonnegative ternary quartic is a sum of three squares. We show more generally that every nonnegative quadratic form on a real projective variety $X$ of minimal degree is a sum of $\dim(X)+1$ squares of linear forms. This strengthens one direction of a recent result due to Blekherman, Smith, and Velasco. Our upper bound is the best possible, and it implies the existence of low-rank factorizations of positive semidefinite bivariate matrix polynomials and representations of biforms as sums of few squares. We determine the number of equivalence classes of sum-of-squares representations of general quadratic forms on surfaces of minimal degree, generalizing the count for ternary quartics by Powers, Reznick, Scheiderer, and Sottile.