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Algebraic boundaries of Hilbert’s SOS cones

Part of: Surfaces and higher-dimensional varieties Nonlinear algebraic or transcendental equations Algorithms - Computer Science Topological rings and modules

Published online by Cambridge University Press:  15 October 2012

Grigoriy Blekherman ,
Jonathan Hauenstein ,
John Christian Ottem ,
Kristian Ranestad  and
Bernd Sturmfels
Grigoriy Blekherman
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (email: greg@math.gatech.edu)
Jonathan Hauenstein
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77845, USA (email: jhauenst@math.tamu.edu)
John Christian Ottem
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB2 1TN, UK (email: J.C.Ottem@dpmms.cam.ac.uk)
Kristian Ranestad
Affiliation:
Department of Mathematics, University of Oslo, 0316 Oslo, Norway (email: ranestad@math.uio.no)
Bernd Sturmfels
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA (email: bernd@math.berkeley.edu)
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Abstract

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We study the geometry underlying the difference between non-negative polynomials and sums of squares (SOS). The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether–Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized, respectively, by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

Keywords

positive polynomialsK3 surfacesnumerical algebraic geometry

MSC classification

Secondary: 13J30: Real algebra 14J28: $K3$ surfaces and Enriques surfaces 14P05: Real algebraic sets 65H10: Systems of equations 68W30: Symbolic computation and algebraic computation
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1717 - 1735
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[BHPS09a]Bates, D., Hauenstein, J., Peterson, C. and Sommese, A., Numerical decomposition of the rank-deficiency set of a matrix of multivariate polynomials, in Approximate commutative algebra, Texts and Monographs in Symbolic Computation, eds Robbiano, L. and Abbott, J. (Springer, Vienna, 2009), 55–77.CrossRefGoogle Scholar
[BHPS09b]Bates, D., Hauenstein, J., Peterson, C. and Sommese, A., A numerical local dimension test for points on the solution set of a system of polynomial equations, SIAM J. Numer. Anal. 47 (2009), 3608–3623.CrossRefGoogle Scholar
[BHSW]Bates, D., Hauenstein, J., Sommese, A. and Wampler, C., Bertini: software for numerical algebraic geometry, http://www.nd.edu/∼sommese/bertini.Google Scholar
[BHSW09]Bates, D., Hauenstein, J., Sommese, A. and Wampler, C., Stepsize control for adaptive multiprecision path tracking, in Interactions of classical and numerical algebraic geometry, Contemporary Mathematics, vol. 496 (American Mathematical Society, Providence, RI, 2009), 21–31.CrossRefGoogle Scholar
[Ble12]Blekherman, G., Non-negative polynomials and sums of squares, J. Amer. Math. Soc. 25 (2012), 617–635.CrossRefGoogle Scholar
[CD05]Cattani, E. and Dickenstein, A., Introduction to residues and resultants, in Solving polynomial equations: foundations, algorithms, and applications, Algorithms and Computation in Mathematics, vol. 14, eds Dickenstein, A. and Emiris, I. Z. (Springer, Berlin, 2005).Google Scholar
[CLR80]Choi, M. D., Lam, T. Y. and Reznick, B., Real zeros of positive semidefinite forms. I, Math. Z. 171 (1980), 1–26.CrossRefGoogle Scholar
[Col93]Colliot-Thélène, J.-L., The Noether–Lefschetz theorem and sums of 4 squares in the rational function field ℝ(x,y), Compositio Math. 86 (1993), 235–243.Google Scholar
[DK07]Dolgachev, I. and Konḏo, S., Moduli of K3 surfaces and complex ball quotients, in Arithmetic and geometry around hypergeometric functions, Progress in Mathematics, vol. 260 (Birkhäuser, Basel, 2007), 43–100.CrossRefGoogle Scholar
[GH85]Griffiths, P. and Harris, J., On the Noether–Lefschetz theorem and some remarks on codimension two cycles, Math. Ann. 271 (1985), 31–51.CrossRefGoogle Scholar
[HT84]Harris, J. and Tu, L., On symmetric and skew-symmetric determinantal varieties, Topology 23 (1984), 71–84.CrossRefGoogle Scholar
[HS10]Hauenstein, J. and Sommese, A., Witness sets of projections, Appl. Math. Comput. 217 (2010), 3349–3354.Google Scholar
[HSW11]Hauenstein, J., Sommese, A. and Wampler, C., Regenerative cascade homotopies for solving polynomial systems, Appl. Math. Comput. 218 (2011), 1240–1246.Google Scholar
[Hil88]Hilbert, D., Über die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann. 32 (1888), 342–350.CrossRefGoogle Scholar
[IK99]Iarrobino, A. and Kanev, V., Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, vol. 1721 (Springer, Berlin, 1999).CrossRefGoogle Scholar
[Jes16]Jessop, C. M., Quartic surfaces with singular points (Cambridge University Press, Cambridge, 1916).Google Scholar
[Kha72]Kharlamov, V. M., The maximal number of components of a fourth degree surface in , Funct. Anal. Appl. 6 (1972), 345–346.CrossRefGoogle Scholar
[KM94]Kontsevich, M. and Manin, Y., Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525–562.CrossRefGoogle Scholar
[Las10]Lasserre, J. B., Moments, positive polynomials and their applications (Imperial College Press, London, 2010).Google Scholar
[Lef21]Lefschetz, S., On certain numerical invariants of algebraic varieties with application to Abelian varieties, Trans. Amer. Math. Soc. 22 (1921), 327–482.CrossRefGoogle Scholar
[MP07]Maulik, D. and Pandharipande, R., Gromov–Witten theory and Noether–Lefschetz theory, Preprint (2007), arXiv:0705.1653.Google Scholar
[Nie12]Nie, J., Discriminants and non-negative polynomials, J. Symbolic Comput. 47 (2012), 167–191.CrossRefGoogle Scholar
[Noe82]Noether, M., Zur Grundlegung der Theorie der algebraischen Raumkurven, J. Reine Angew. Math. 92 (1882), 271–318.CrossRefGoogle Scholar
[Ott07]Ottaviani, G., Symplectic bundles on the plane, secant varieties and Lüroth quartics revisited, in Vector bundles and low-codimensional subvarieties: state of the art and recent developments, Quaderni di Matematica, vol. 21 (Dipartimento di Matematica, Seconda Università degli Studi di Napoli, Caserta, 2007), 315–352.Google Scholar
[PS71]Pjateckiǐ-Šapiro, I. and Šafarevič, I. R., A Torelli theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572.Google Scholar
[PSV12]Plaumann, D., Sturmfels, B. and Vinzant, C., Computing linear matrix representations of Helton–Vinnikov curves, in Mathematical methods in systems, optimization and control (volume dedicated to Bill Helton, eds H. Dym, M. de Oliveira and M. Putinar), Operator Theory: Advances and Applications, vol. 222 (Birkhäuser, Basel, 2012), 259–277.CrossRefGoogle Scholar
[Rez07]Reznick, B., On Hilbert’s construction of positive polynomials, Preprint (2007), arXiv:0707.2156.Google Scholar
[Roh13]Rohn, K., Die Maximalzahl und Anordnung der Ovale bei der ebenen Kurve 6. Ordnung und bei der Fläche 4. Ordnung, Math. Ann. 73 (1913), 177–228.CrossRefGoogle Scholar
[RS10]Rostalski, P. and Sturmfels, B., Dualities in convex algebraic geometry, Rend. Mat. Appl. (7) 30 (2010), 285–327.Google Scholar
[Sai74]Saint-Donat, B., Projective models of K3 surfaces, Amer. J. Math. 96 (1974), 602–639.CrossRefGoogle Scholar
[Sch93]Schneider, R., Convex bodies: the Brunn–Minkowski theory (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
[SV00]Sommese, A. and Verschelde, J., Numerical homotopies to compute generic points on positive dimensional algebraic sets, J. Complexity 16 (2000), 572–602.CrossRefGoogle Scholar
[SW05]Sommese, A. and Wampler, C., The numerical solution of systems of polynomials arising in engineering and science (World Scientific, Singapore, 2005).CrossRefGoogle Scholar
[VC93]Verschelde, J. and Cools, R., Symbolic homotopy construction, Appl. Algebra Engrg. Comm. Comput. 4 (1993), 169–183.CrossRefGoogle Scholar