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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Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36. Springer, Berlin (1998). Translated from the 1987 French original; Revised by the authors
Dey, P., Pillai, H.K.: A complete characterization of determinantal quadratic polynomials. Linear Algebra Appl. 543, 106–124 (2018)
Dey, P.: Definite Determinantal Representations via Orthostochastic Matrices. (2019). arXiv:1708.09559
Dey, P.: Definite determinantal representations of multivariate polynomials. J. Algebra Appl. 2050129 (2020)
Gelfand, I.M., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser, Boston (1994)
Gondard, D., Ribenboim, P.: Le 17e probleme de Hilbert pour les matrices. Bull. Sci. Math. 2(98), 1 (1974)
Grayson, D.R., Stillman, M.E.: Macaulay2, a Software System for Research in Algebraic Geometry (2002)
Grinshpan, A., Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V., Woerdeman, H.J.: Stable and real-zero polynomials in two variables. Multidimens. Syst. Signal Process. 27(1), 1–26 (2014)
Henrion, D.: Detecting rigid convexity of bivariate polynomials. Linear Algebra Appl. 432(5), 1218–1233 (2010)
Helton, J.W., Vinnikov, V.: Linear matrix inequality representation of sets. Commun. Pure Appl. Math. 60(5), 654–674 (2007)
Hörmander, L.: Linear Partial Differential Operators, Die Grundlehren der mathematischen Wissenschaften. Academic Press Inc., Publishers, Springer, New York, Berlin (1963)
Krein, M.G., Naimark, M.A.: The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations. Linear Multilinear Algebra 10(4), 265–308 (1981)
Lombardi, H., Perrucci, D., Roy, M.-F.: An elementary recursive bound for effective Positivstellensatz and Hilbert 17-th problem. p. 113. (2020)
Leykin, A., Plaumann, D.: Determinantal representations of hyperbolic curves via polynomial homotopy continuation. Math. Comput. 86(308), 2877–2888 (2017)
Marshall, M.: Positive polynomials and sums of squares. In: Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence (2008)
Marshall, M.: Polynomials non-negative on a strip. Proc. Am. Math. Soc. 138(5), 1559–1567 (2010)
Netzer, T., Plaumann, D., Thom, A.: Determinantal representations and the Hermite matrix. Mich. Math. J. 62(2), 407–420 (2013)
Netzer, T., Thom, A.: Polynomials with and without determinantal representations. Linear Algebra Appl. 437(7), 1579–1595 (2012)
Nuij, W.: A note on hyperbolic polynomials. Math. Scand. 23(1968), 69–72 (1969)
Peyrl, H., Parrilo, P.A.: Computing sum of squares decompositions with rational coefficients. Theor. Comput. Sci. 409(2), 269–281 (2008)
Plaumann, D., Sturmfels, B., Vinzant, C.: Computing linear matrix representations of Helton–Vinnikov curves. In: Mathematical Methods in Systems, Optimization, and Control. Oper. Theory Adv. Appl., vol. 222, pp. 259–277. Birkhäuser/Springer Basel AG, Basel (2012)
Saunderson, J.: Certifying Polynomial Nonnegativity via Hyperbolic Optimization. arXiv:1904.00491 (2019)
Raghavendra, P., Ryder, N., Srivastava, N.: Real stability testing. In: 8th Innovations in Theoretical Computer Science Conference, LIPIcs. Leibniz Int. Proc. Inform., vol. 67, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, Art. No. 5, 15 (2017)
Vinnikov, V.: LMI representations of convex semialgebraic sets and determinantal representations of algebraic hypersurfaces: past, present, and future. Oper. Theory: Adv. Appl. 222, 325–348 (2012)
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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Dey, P., Plaumann, D. Testing Hyperbolicity of Real Polynomials. Math.Comput.Sci. 14, 111–121 (2020). https://doi.org/10.1007/s11786-019-00449-w
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DOI: https://doi.org/10.1007/s11786-019-00449-w