A convex form that is not a sum of squares

TL;DR

This paper constructs an explicit example of a convex degree-four form in 272 variables that is not a sum of squares, advancing understanding of convex polynomials outside sum-of-squares representations.

Contribution

It provides the first explicit example of a convex form not representable as a sum of squares, using symmetry and connections to the Cauchy-Schwarz inequality over octonions.

Findings

Explicit example of a convex form not sum of squares in 272 variables

Improved bounds on sum-of-squares relaxation for quaternary quartic forms

Connection between convexity and near-constant forms on the sphere

Abstract

Every convex homogeneous polynomial (or form) is nonnegative. Blekherman has shown that there exist convex forms that are not sums of squares via a nonconstructive argument. We provide an explicit example of a convex form of degree four in 272 variables that is not a sum of squares. The form is related to the Cauchy-Schwarz inequality over the octonions. The proof uses symmetry reduction together with the fact (due to Blekherman) that forms of even degree, that are near-constant on the unit sphere, are convex. Using this same connection, we obtain improved bounds on the approximation quality achieved by the basic sum-of-squares relaxation for optimizing quaternary quartic forms on the sphere.