The analogue of Hilbert's 1888 theorem for Even 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 $(n,2d)=(n,4{)}_{n\ge 3}$ or $(n,2d)=(3,8)$.

11 pages, 1 figure. arXiv admin note: text overlap with arXiv:1505.08145

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