On the Choi–Lam analogue of Hilbert's 1888 theorem for symmetric forms

A famous theorem of Hilbert from 1888 states that for given n and d, every positive semidefinite (psd) real form of degree 2d in n variables is a sum of squares (sos) of real forms if and only if n=2 or d=1 or (n,2d)=(3,4). In 1976, Choi and Lam proved the analogue of Hilbert's Theorem for symm...

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Bibliographic Details
Published in:Linear algebra and its applications 2016-05, Vol.496, p.114-120
Main Authors: Goel, Charu, Kuhlmann, Salma, Reznick, Bruce
Format: Article
Language:English
Subjects:
Positive polynomials
Sums of squares
Symmetric forms
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Summary:A famous theorem of Hilbert from 1888 states that for given n and d, every positive semidefinite (psd) real form of degree 2d in n variables is a sum of squares (sos) of real forms if and only if n=2 or d=1 or (n,2d)=(3,4). 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≥5. In this paper we complete their proof by constructing explicit psd not sos symmetric n-ary quartics for n≥5.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2016.01.024