Sums of squares and varieties of minimal degree

Blekherman, Grigoriy; Smith, Gregory; Velasco, Mauricio

doi:10.1090/jams/847

Sums of squares and varieties of minimal degree
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by Grigoriy Blekherman, Gregory G. Smith and Mauricio Velasco

J. Amer. Math. Soc. 29 (2016), 893-913

DOI: https://doi.org/10.1090/jams/847

Published electronically: September 3, 2015

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Abstract:

Let $X \subseteq \mathbb {P}^n$ be a real nondegenerate subvariety such that the set $X(\mathbb {R})$ of real points is Zariski dense. We prove that every real quadratic form that is nonnegative on $X(\mathbb {R})$ is a sum of squares of linear forms if and only if $X$ is a variety of minimal degree. This substantially extends Hilbert’s celebrated characterization of equality between nonnegative forms and sums of squares. We obtain a complete list for the cases of equality and also a classification of the lattice polytopes $Q$ for which every nonnegative Laurent polynomial with support contained in $2Q$ is a sum of squares.

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