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Positive Gorenstein ideals

Author: Grigoriy Blekherman
Journal: Proc. Amer. Math. Soc. 143 (2015), 69-86
MSC (2010): Primary 14N05, 14P99; Secondary 90C22, 47A57, 52A20
Published electronically: August 29, 2014
MathSciNet review: 3272733
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Abstract: We introduce positive Gorenstein ideals. These are Gorenstein ideals in the graded ring $ \mathbb{R}[x]$ with socle in degree $ 2d$ which when viewed as a linear functional on $ \mathbb{R}[x]_{2d}$ is nonnegative on squares. Equivalently, positive Gorenstein ideals are apolar ideals of forms whose differential operator is nonnegative on squares. Positive Gorenstein ideals arise naturally in the context of nonnegative polynomials and sums of squares, and they provide a powerful framework for studying concrete aspects of sums of squares representations. We present applications of positive Gorenstein ideals in real algebraic geometry, analysis and optimization. In particular, we present a simple proof of Hilbert's nearly forgotten result on representations of ternary nonnegative forms as sums of squares of rational functions. Drawing on our previous work (2012), our main tools are Cayley-Bacharach duality and elementary convex geometry.

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  • [1] Grigoriy Blekherman, Nonnegative polynomials and sums of squares, J. Amer. Math. Soc. 25 (2012), no. 3, 617-635. MR 2904568,
  • [2] Grigoriy Blekherman, Jonathan Hauenstein, John Christian Ottem, Kristian Ranestad, and Bernd Sturmfels, Algebraic boundaries of Hilbert's SOS cones, Compos. Math. 148 (2012), no. 6, 1717-1735. MR 2999301,
  • [3] Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998. Translated from the 1987 French original; revised by the authors. MR 1659509 (2000a:14067)
  • [4] Mats Boij, Enrico Carlini, and Anthony V. Geramita, Monomials as sums of powers: the real binary case, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3039-3043. MR 2811260 (2012e:11070),
  • [5] Pierre Comon and Giorgio Ottaviani, On the typical rank of real binary forms, Linear Multilinear Algebra 60 (2012), no. 6, 657-667. MR 2929176,
  • [6] Enrico Carlini, Maria Virginia Catalisano, and Anthony V. Geramita, The solution to the Waring problem for monomials and the sum of coprime monomials, J. Algebra 370 (2012), 5-14. MR 2966824,
  • [7] David Eisenbud, Mark Green, and Joe Harris, Cayley-Bacharach theorems and conjectures, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 3, 295-324. MR 1376653 (97a:14059),
  • [8] Joe Harris, Algebraic geometry: A first course, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. MR 1182558 (93j:14001)
  • [9] David Hilbert, Ueber die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann. 32 (1888), no. 3, 342-350 (German). MR 1510517,
  • [10] David Hilbert, Über ternäre definite Formen, Acta Math. 17 (1893), no. 1, 169-197 (German). MR 1554835,
  • [11] Anthony Iarrobino and Vassil Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, vol. 1721, Springer-Verlag, Berlin, 1999. Appendix C by Iarrobino and Steven L. Kleiman. MR 1735271 (2001d:14056)
  • [12] Jiawang Nie and Markus Schweighofer, On the complexity of Putinar's Positivstellensatz, J. Complexity 23 (2007), no. 1, 135-150. MR 2297019 (2008b:14095),
  • [13] J. M. Landsberg and Zach Teitler, On the ranks and border ranks of symmetric tensors, Found. Comput. Math. 10 (2010), no. 3, 339-366. MR 2628829 (2011d:14095),
  • [14] Jean B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim. 11 (2000/01), no. 3, 796-817. MR 1814045 (2002b:90054),
  • [15] Jean Bernard Lasserre, Moments, positive polynomials and their applications, Imperial College Press Optimization Series, vol. 1, Imperial College Press, London, 2010. MR 2589247 (2011c:90001)
  • [16] Juan Migliore, Uwe Nagel, and Fabrizio Zanello, On the degree two entry of a Gorenstein $ h$-vector and a conjecture of Stanley, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2755-2762. MR 2399039 (2009b:13038),
  • [17] Pablo A. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Algebraic and geometric methods in discrete optimization, Math. Program. 96 (2003), no. 2, Ser. B, 293-320. MR 1993050 (2004g:90075),
  • [18] Victoria Powers and Thorsten Wörmann, An algorithm for sums of squares of real polynomials, J. Pure Appl. Algebra 127 (1998), no. 1, 99-104. MR 1609496 (99a:11047),
  • [19] Bruce Reznick, Sums of even powers of real linear forms, Mem. Amer. Math. Soc. 96 (1992), no. 463, viii+155. MR 1096187 (93h:11043)
  • [20] 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 (2001i:11042),
  • [21] Raman Sanyal, Frank Sottile, and Bernd Sturmfels, Orbitopes, Mathematika 57 (2011), no. 2, 275-314. MR 2825238 (2012g:52001),

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Additional Information

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

Received by editor(s): May 22, 2012
Received by editor(s) in revised form: May 23, 2012, and March 8, 2013
Published electronically: August 29, 2014
Communicated by: Harm Derksen
Article copyright: © Copyright 2014 American Mathematical Society

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