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

Charu Goel; Salma Kuhlmann; Bruce Reznick

doi:10.1016/j.jpaa.2016.10.003

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arXiv · 1509.07482

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

Charu Goel·Salma Kuhlmann·Bruce Reznick

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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 $(n,2d)= (3,8)$.

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Charu Goel, Salma Kuhlmann, Bruce Reznick. 2015-09-24. The analogue of Hilbert's 1888 theorem for Even Symmetric Forms. https://doi.org/10.1016/j.jpaa.2016.10.003

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