Sums of squares and varieties of minimal degree

Blekherman, Grigoriy¹ ; Smith, Gregory² ; Velasco, Mauricio³

¹ School of Mathematics, Georgia Tech, 686 Cherry Street, Atlanta, Georgia, 30332
² Department of Mathematics & Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada
³ Departamento de Matemáticas, Universidad de los Andes, Carrera 1 No. 18a 10, Edificio H, Primer Piso, 111711 Bogotá, Colombia

Journal of the American Mathematical Society, Tome 29 (2016) no. 3, pp. 893-913

Résumé

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.

Détail

DOI : 10.1090/jams/847

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