The analogue of Hilbert's 1888 theorem for Even Symmetric Forms

TL;DR

This paper extends Hilbert's 1888 theorem to even symmetric forms, characterizing when positive semidefinite forms are sums of squares based on variables and degree.

Contribution

It provides a complete characterization of when even symmetric psd forms are sos, generalizing Hilbert's classical result to this specific class.

Findings

An even symmetric n-ary 2d-ic psd form is sos if and only if n=2, d=1, (n,2d)=(n,4) for n≥3, or (3,8).

The result fully classifies sos conditions for even symmetric forms.

The theorem extends Hilbert's original classification to a broader class of symmetric forms.

Abstract

Hilbert proved in 1888 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. We study the analogue for even symmetric forms. We establish that an even symmetric $n$-ary $2d$-ic psd form is sos if and only if $n=2$ or $d=1$ or $(n,2d)=(n,4)_{n \geq 3}$ or $(n,2d)= (3,8)$.