Nonnegative polynomials and sums of squares

Author:
Grigoriy Blekherman

Journal:
J. Amer. Math. Soc. **25** (2012), 617-635

MSC (2010):
Primary 14N05, 14P99; Secondary 52A20

DOI:
https://doi.org/10.1090/S0894-0347-2012-00733-4

Published electronically:
March 15, 2012

MathSciNet review:
2904568

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Abstract: In the smallest cases where there exist nonnegative polynomials that are not sums of squares we present a complete explanation of this distinction. The fundamental reason that the cone of sums of squares is strictly contained in the cone of nonnegative polynomials is that polynomials of degree satisfy certain linear relations, known as the Cayley-Bacharach relations, which are not satisfied by polynomials of full degree . For any nonnegative polynomial that is not a sum of squares we can write down a linear inequality coming from a Cayley-Bacharach relation that certifies this fact. We also characterize strictly positive sums of squares that lie on the boundary of the cone of sums of squares and extreme rays of the cone dual to the cone of sums of squares.

**1.**G. Blekherman,*There are significantly more nonnegative polynomials than sums of squares*, Israel J. of Math., vol. 183, 355-380, 2006. MR**2254649 (2007f:14062)****2.**G. Blekherman, J. Hauenstein, J. C. Ottem, K. Ranestad, B. Sturmfels,*Algebraic Boundaries of Hilbert's SOS Cones*, submitted for publication, arXiv:1107.1846.**3.**J. Bochnak, M. Coste, M.-F. Roy,*Real Algebraic Geometry,*Springer-Verlag, Berlin, 1998. MR**1659509 (2000a:14067)****4.**E. Cattani and A. Dickenstein, Introduction to residues and resultants, in*Solving Polynomial Equations: Foundations, Algorithms, and Applications*(eds., A. Dickenstein and I. Z. Emiris), Algorithms and Computation in Mathematics**14**, Springer, 2005. MR**2161984 (2008d:14095)****5.**M. D. Choi, T. Y. Lam, B. Reznick*Even symmetric sextics.*Math. Z. 195, no. 4, 559-580, 1987. MR**900345 (88j:11019)****6.**D. Eisenbud, M. Green, J. Harris,*Cayley-Bacharach theorems and conjectures*, Bull. Amer. Math. Soc., vol. 33, no. 3, 295-324, 1996. MR**1376653 (97a:14059)****7.**G. Hardy, E. Littlewood, G. Polya,*Inequalities,*Cambridge University Press, Cambridge, 1988. MR**944909 (89d:26016)****8.**J. Harris,*Algebraic Geometry. A First Course,*Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1995. MR**1416564 (97e:14001)****9.**J. Nie, M. Schweighofer,*On the complexity of Putinar's Positivstellensatz,*J. of Complex., vol. 23, no. 1, 135-150, 2007. MR**2297019 (2008b:14095)****10.**J. B. Lasserre,*Global optimization with polynomials and the problem of moments,*SIAM J. Optim., vol. 11, no. 3, 796-817 (electronic), 2000/01. MR**1814045 (2002b:90054)****11.**P. Parrilo,*Semidefinite programming relaxations for semialgebraic problems,*Math. Program., vol. 96, no. 2, Ser. B, 293-320, 2000/01. MR**1993050 (2004g:90075)****12.**M. Ramana, A. J. Goldman,*Some geometric results in semidefinite programming*, J. Global Optim., vol. 7, no. 1, 33-50, 1995. MR**1342934 (96i:90059)****13.**B. Reznick,*Sums of Even Powers of Real Linear Forms*, Mem. Amer. Math. Soc., vol. 96, no. 463, 1992. MR**1096187 (93h:11043)****14.**B. Reznick,*Some concrete aspects of Hilbert's 17th Problem*, Contemp. Math., no. 253, 251-272, 2000. MR**1747589 (2001i:11042)****15.**B.Reznick,*On Hilbert's construction of positive polynomials,*arXiv:0707.2156.**16.**R. Sanyal, F. Sottile, B. Sturmfels,*Orbitopes,*Mathematika, vol. 57 (2011), 275-314. MR**2825238****17.**R. Schneider,*Convex Bodies: the Brunn-Minkowski Theory,*Cambridge University Press, Cambridge, 1993. MR**1216521 (94d:52007)**

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Additional Information

**Grigoriy Blekherman**

Affiliation:
School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332-0160

Email:
greg@math.gatech.edu

DOI:
https://doi.org/10.1090/S0894-0347-2012-00733-4

Received by editor(s):
December 11, 2010

Received by editor(s) in revised form:
August 12, 2011, and December 17, 2011

Published electronically:
March 15, 2012

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.