On the Choi-Lam analogue of Hilbert's 1888 theorem for Symmetric Forms

TL;DR

This paper completes the proof of a symmetric form analogue of Hilbert's 1888 theorem by explicitly constructing positive semidefinite but not sum of squares symmetric quartic forms for all n ≥ 5.

Contribution

It provides explicit constructions of psd not sos symmetric quartic forms for all n ≥ 5, completing the previous theoretical proof.

Findings

Explicit psd not sos symmetric quartic forms for n ≥ 5

Completes the proof of the symmetric analogue of Hilbert's theorem

Extends the class of known examples in polynomial positivity

Abstract

A famous theorem of Hilbert from 1888 states that a positive semidefinite (psd) real form is a sum of squares (sos) of real forms if and only if $n=2$ or $d=1$ or $(n,2d)=(3,4)$, where $n$ is the number of variables and $2d$ the degree of the form. In 1976, Choi and Lam proved the analogue of Hilbert's Theorem for symmetric forms by assuming the existence of psd not sos symmetric $n$-ary quartics for $n \geq 5$. In this paper we complete their proof by constructing explicit psd not sos symmetric $n$-ary quartics for $n \geq 5$.