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Algebraic boundaries of Hilbert’s SOS cones

Published online by Cambridge University Press:  15 October 2012

Grigoriy Blekherman
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (email: [email protected])
Jonathan Hauenstein
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77845, USA (email: [email protected])
John Christian Ottem
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB2 1TN, UK (email: [email protected])
Kristian Ranestad
Affiliation:
Department of Mathematics, University of Oslo, 0316 Oslo, Norway (email: [email protected])
Bernd Sturmfels
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA (email: [email protected])
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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.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2012

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