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arXiv · 1308.0751

Sums of squares and varieties of minimal degree

Grigoriy Blekherman·Gregory G. Smith·Mauricio Velasco
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Abstract

Let X be a real nondegenerate projective subvariety such that its set of real points is Zariski dense. We prove that every real quadratic form that is nonnegative on X 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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Grigoriy Blekherman, Gregory G. Smith, Mauricio Velasco. 2013-08-03. Sums of squares and varieties of minimal degree. https://doi.org/10.1090/jams/847

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