On Sum of Squares Representation of Convex Forms and Generalized Cauchy-Schwarz Inequalities

TL;DR

This paper proves that all convex forms in four variables of degree four are sums of squares and explores conditions under which convex forms in fewer variables and higher degrees are also sums of squares, introducing generalized inequalities.

Contribution

It establishes the sum of squares property for convex forms in four variables of degree four and links this to a conjecture for higher degrees and fewer variables, advancing understanding of convex form representations.

Findings

All convex forms in 4 variables of degree 4 are sums of squares.

Conditional proof that convex forms in 3 variables of degree 6 are sums of squares, assuming Blekherman's conjecture.

Introduction of generalized Cauchy-Schwarz inequalities related to convex forms.

Abstract

A convex form of degree larger than one is always nonnegative since it vanishes together with its gradient at the origin. In 2007, Parrilo asked if convex forms are always sums of squares. A few years later, Blekherman answered the question in the negative by showing through volume arguments that for high enough number of variables, there must be convex forms of degree as low as 4 that are not sums of squares. Remarkably, no examples are known to date. In this paper, we show that all convex forms in 4 variables and of degree 4 are sums of squares. We also show that if a conjecture of Blekherman related to the so-called Cayley-Bacharach relations is true, then the same statement holds for convex forms in 3 variables and of degree 6. These are the two minimal cases where one would have any hope of seeing convex forms that are not sums of squares (due to known obstructions). A main ingredient of the proof is the derivation of certain "generalized Cauchy-Schwarz inequalities" which could be of independent interest.