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Sums of squares and varieties of minimal degree


Authors: Grigoriy Blekherman, Gregory G. Smith and Mauricio Velasco
Journal: J. Amer. Math. Soc. 29 (2016), 893-913
MSC (2010): Primary 14P05; Secondary 12D15, 90C22
DOI: https://doi.org/10.1090/jams/847
Published electronically: September 3, 2015
MathSciNet review: 3486176
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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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Additional Information

Grigoriy Blekherman
Affiliation: School of Mathematics, Georgia Tech, 686 Cherry Street, Atlanta, Georgia, 30332
Email: greg@math.gatech.edu

Gregory G. Smith
Affiliation: Department of Mathematics & Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada
Email: ggsmith@mast.queensu.ca

Mauricio Velasco
Affiliation: Departamento de Matemáticas, Universidad de los Andes, Carrera 1 No. 18a 10, Edificio H, Primer Piso, 111711 Bogotá, Colombia
Email: mvelasco@uniandes.edu.co

DOI: https://doi.org/10.1090/jams/847
Received by editor(s): January 7, 2014
Received by editor(s) in revised form: May 8, 2015, and July 23, 2015
Published electronically: September 3, 2015
Additional Notes: The first author was supported in part by a Sloan Fellowship, NSF Grant DMS-0757212, the Mittag-Leffler Institute, and IPAM
The second author was supported in part by NSERC, the Mittag-Leffler Institute, and MSRI
The third author was supported in part by the FAPA grants from Universidad de los Andes
Article copyright: © Copyright 2015 American Mathematical Society