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

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 $n\ge 5$. In this paper we complete their proof by constructing explicit psd not sos symmetric $n$-ary quartics for $n\ge 5$.

The previous submission has been improved and split into two papers. The first one being the present version and the second one called "The analogue of Hilbert's 1888 theorem for Even Symmetric Forms" in which we completed our conjecture, namely, an even symmetric n-ary 2d-ic psd form is sos if and only if n=2 or d=1 or (n,2d)=(3,8) or (n,2d)=(n,4) for n greater than or equal to 3