## 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.## References

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## Bibliographic Information

**Grigoriy Blekherman**- Affiliation: School of Mathematics, Georgia Tech, 686 Cherry Street, Atlanta, Georgia, 30332
- MR Author ID: 668861
- Email: greg@math.gatech.edu
**Gregory G. Smith**- Affiliation: Department of Mathematics & Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada
- MR Author ID: 622959
- 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
- 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 - © Copyright 2015 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**29**(2016), 893-913 - MSC (2010): Primary 14P05; Secondary 12D15, 90C22
- DOI: https://doi.org/10.1090/jams/847
- MathSciNet review: 3486176