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.

References

Victor Batyrev and Benjamin Nill, Multiples of lattice polytopes without interior lattice points, Mosc. Math. J. 7 (2007), no. 2, 195–207, 349 (English, with English and Russian summaries). MR 2337878, DOI 10.17323/1609-4514-2007-7-2-195-207
Matthias Beck and Sinai Robins, Computing the continuous discretely, Undergraduate Texts in Mathematics, Springer, New York, 2007. Integer-point enumeration in polyhedra. MR 2271992
Eberhard Becker, Valuations and real places in the theory of formally real fields, Real algebraic geometry and quadratic forms (Rennes, 1981) Lecture Notes in Math., vol. 959, Springer, Berlin-New York, 1982, pp. 1–40. MR 683127
Grigoriy Blekherman, Nonnegative polynomials and sums of squares, J. Amer. Math. Soc. 25 (2012), no. 3, 617–635. MR 2904568, DOI 10.1090/S0894-0347-2012-00733-4
Grigoriy Blekherman, Pablo A. Parrilo, and Rekha R. Thomas (eds.), Semidefinite optimization and convex algebraic geometry, MOS-SIAM Series on Optimization, vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2013. MR 3075433
Grigoriy Blekherman, Sadik Iliman, and Martina Kubitzker, Dimensional differences between faces of the cones of nonnegative polynomials and sums of squares, Int. Math. Res. Not. IMRN., DOI 10.1093/imrn/rnu202
Markus Brodmann and Peter Schenzel, Arithmetic properties of projective varieties of almost minimal degree, J. Algebraic Geom. 16 (2007), no. 2, 347–400. MR 2274517, DOI 10.1090/S1056-3911-06-00442-5
David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322, DOI 10.1090/gsm/124
Man Duen Choi, Tsit Yuen Lam, and Bruce Reznick, Real zeros of positive semidefinite forms. I, Math. Z. 171 (1980), no. 1, 1–26. MR 566480, DOI 10.1007/BF01215051
David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89–133. MR 741934, DOI 10.1016/0021-8693(84)90092-9
David Eisenbud and Joe Harris, On varieties of minimal degree (a centennial account), Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 3–13. MR 927946, DOI 10.1090/pspum/046.1/927946
Joseph Gubeladze, The isomorphism problem for commutative monoid rings, J. Pure Appl. Algebra 129 (1998), no. 1, 35–65. MR 1626643, DOI 10.1016/S0022-4049(97)00063-7
Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
Akihiro Higashitani, Counterexamples of the conjecture on roots of Ehrhart polynomials, Discrete Comput. Geom. 47 (2012), no. 3, 618–623. MR 2891252, DOI 10.1007/s00454-011-9390-4
David Hilbert, Ueber die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann. 32 (1888), no. 3, 342–350 (German). MR 1510517, DOI 10.1007/BF01443605
Jean-Pierre Jouanolou, Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42, Birkhäuser Boston, Inc., Boston, MA, 1983 (French). MR 725671
Jean Bernard Lasserre, Moments, positive polynomials and their applications, Imperial College Press Optimization Series, vol. 1, Imperial College Press, London, 2010. MR 2589247
Monique Laurent, Sums of squares, moment matrices and optimization over polynomials, Emerging applications of algebraic geometry, IMA Vol. Math. Appl., vol. 149, Springer, New York, 2009, pp. 157–270. MR 2500468, DOI 10.1007/978-0-387-09686-5_{7}
S. L′vovsky, On inflection points, monomial curves, and hypersurfaces containing projective curves, Math. Ann. 306 (1996), no. 4, 719–735. MR 1418349, DOI 10.1007/BF01445273
Motakuri Ramana and A. J. Goldman, Some geometric results in semidefinite programming, J. Global Optim. 7 (1995), no. 1, 33–50. MR 1342934, DOI 10.1007/BF01100204
Bruce Reznick, Some concrete aspects of Hilbert’s 17th Problem, Real algebraic geometry and ordered structures (Baton Rouge, LA, 1996) Contemp. Math., vol. 253, Amer. Math. Soc., Providence, RI, 2000, pp. 251–272. MR 1747589, DOI 10.1090/conm/253/03936
Claus Scheiderer, A Positivstellensatz for projective real varieties, Manuscripta Math. 138 (2012), no. 1-2, 73–88. MR 2898748, DOI 10.1007/s00229-011-0484-3
Richard P. Stanley, A monotonicity property of $h$-vectors and $h^*$-vectors, European J. Combin. 14 (1993), no. 3, 251–258. MR 1215335, DOI 10.1006/eujc.1993.1028
A. Stapledon, Inequalities and Ehrhart $\delta$-vectors, Trans. Amer. Math. Soc. 361 (2009), no. 10, 5615–5626. MR 2515826, DOI 10.1090/S0002-9947-09-04776-X
F. L. Zak, Projective invariants of quadratic embeddings, Math. Ann. 313 (1999), no. 3, 507–545. MR 1678545, DOI 10.1007/s002080050271