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Testing Hyperbolicity of Real Polynomials

Abstract

Hyperbolic polynomials are real multivariate polynomials with only real roots along a fixed pencil of lines. Testing whether a given polynomial is hyperbolic is a difficult task in general. We examine different ways of translating hyperbolicity into nonnegativity conditions, which can then be tested via sum-of-squares relaxations.

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Acknowledgements

We would like to thank Amir Ali Ahmadi, Diego Cifuentes and especially Elias Tsigaridas for helpful discussions on the subject of this paper. We also thank the referees for their careful reading and useful comments. Much of the work on this paper has been supported by the National Science Foundation under Grant No. DMS-1439786 while both authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Fall 2018 Nonlinear Algebra program. The first author also gratefully acknowledges support through the Max Planck Institute for Mathematics in the Sciences in Leipzig.

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Correspondence to Papri Dey.

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Dey, P., Plaumann, D. Testing Hyperbolicity of Real Polynomials. Math.Comput.Sci. (2020) doi:10.1007/s11786-019-00449-w

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Keywords

  • Hyperbolic polynomials
  • Real zero polynomials
  • Real Nullstellensatz
  • Resultants

Mathematics Subject Classification

  • 13P05
  • 13P15
  • 13P25
  • 26C10
  • 34C08
  • 68W30