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Algebraic boundaries of Hilbert’s SOS cones

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

Information

Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1717 - 1735
Copyright
Copyright © Foundation Compositio Mathematica 2012

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