Roots of Gårding hyperbolic polynomials

Rainer, Armin

doi:10.1090/proc/15634

Roots of Gårding hyperbolic polynomials
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Proc. Amer. Math. Soc. 150 (2022), 2433-2446 Request permission

Abstract:

We explore the regularity of the roots of Gårding hyperbolic polynomials and real stable polynomials. As an application we obtain new regularity results of Sobolev type for the eigenvalues of Hermitian matrices and for the singular values of arbitrary matrices. These results are optimal among all Sobolev spaces.

References

Dmitri Alekseevsky, Andreas Kriegl, Peter W. Michor, and Mark Losik, Choosing roots of polynomials smoothly, Israel J. Math. 105 (1998), 203–233. MR 1639759, DOI 10.1007/BF02780330
Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
M. F. Atiyah, R. Bott, and L. Gȧrding, Lacunas for hyperbolic differential operators with constant coefficients. I, Acta Math. 124 (1970), 109–189. MR 470499, DOI 10.1007/BF02394570
Miroslav Bačák and Jonathan M. Borwein, On difference convexity of locally Lipschitz functions, Optimization 60 (2011), no. 8-9, 961–978. MR 2860286, DOI 10.1080/02331931003770411
Heinz H. Bauschke, Osman Güler, Adrian S. Lewis, and Hristo S. Sendov, Hyperbolic polynomials and convex analysis, Canad. J. Math. 53 (2001), no. 3, 470–488. MR 1827817, DOI 10.4153/CJM-2001-020-6
Jean-Michel Bony, Fabrizio Broglia, Ferruccio Colombini, and Ludovico Pernazza, Nonnegative functions as squares or sums of squares, J. Funct. Anal. 232 (2006), no. 1, 137–147. MR 2200169, DOI 10.1016/j.jfa.2005.06.011
J. Borcea and P. Brändén, Multivariate Pólya-Schur classification problems in the Weyl algebra, Proc. Lond. Math. Soc. (3) 101 (2010), no. 1, 73–104. MR 2661242, DOI 10.1112/plms/pdp049
Gérard Bourdaud, Le calcul fonctionnel dans les espaces de Sobolev, Invent. Math. 104 (1991), no. 2, 435–446 (French). MR 1098617, DOI 10.1007/BF01245083
M. D. Bronšteĭn, Smoothness of roots of polynomials depending on parameters, Sibirsk. Mat. Zh. 20 (1979), no. 3, 493–501, 690 (Russian). MR 537355
Ferruccio Colombini, Nicola Orrù, and Ludovico Pernazza, On the regularity of the roots of hyperbolic polynomials, Israel J. Math. 191 (2012), no. 2, 923–944. MR 3011501, DOI 10.1007/s11856-012-0015-2
R. M. Dudley, On second derivatives of convex functions, Math. Scand. 41 (1977), no. 1, 159–174. MR 482164, DOI 10.7146/math.scand.a-11710
Lars Gårding, Linear hyperbolic partial differential equations with constant coefficients, Acta Math. 85 (1951), 1–62. MR 41336, DOI 10.1007/BF02395740
Lars Gȧrding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957–965. MR 0113978, DOI 10.1512/iumj.1959.8.58061
Leonid Gurvits, Van der Waerden/Schrijver-Valiant like conjectures and stable (aka hyperbolic) homogeneous polynomials: one theorem for all, Electron. J. Combin. 15 (2008), no. 1, Research Paper 66, 26. With a corrigendum. MR 2411443
Philip Hartman, On functions representable as a difference of convex functions, Pacific J. Math. 9 (1959), 707–713. MR 110773, DOI 10.2140/pjm.1959.9.707
F. Reese Harvey and H. Blaine Lawson Jr., Gårding’s theory of hyperbolic polynomials, Comm. Pure Appl. Math. 66 (2013), no. 7, 1102–1128. MR 3055586, DOI 10.1002/cpa.21443
J. William Helton and Victor Vinnikov, Linear matrix inequality representation of sets, Comm. Pure Appl. Math. 60 (2007), no. 5, 654–674. MR 2292953, DOI 10.1002/cpa.20155
J.-B. Hiriart-Urruty, Generalized differentiability, duality and optimization for problems dealing with differences of convex functions, Convexity and duality in optimization (Groningen, 1984) Lecture Notes in Econom. and Math. Systems, vol. 256, Springer, Berlin, 1985, pp. 37–70. MR 873269, DOI 10.1007/978-3-642-45610-7_{3}
Andreas Kriegl and Peter W. Michor, Differentiable perturbation of unbounded operators, Math. Ann. 327 (2003), no. 1, 191–201. MR 2006008, DOI 10.1007/s00208-003-0446-5
Krzysztof Kurdyka and Laurentiu Paunescu, Hyperbolic polynomials and multiparameter real-analytic perturbation theory, Duke Math. J. 141 (2008), no. 1, 123–149. MR 2372149, DOI 10.1215/S0012-7094-08-14113-4
P. D. Lax, Differential equations, difference equations and matrix theory, Comm. Pure Appl. Math. 11 (1958), 175–194. MR 98110, DOI 10.1002/cpa.3160110203
Giovanni Leoni, A first course in Sobolev spaces, Graduate Studies in Mathematics, vol. 105, American Mathematical Society, Providence, RI, 2009. MR 2527916, DOI 10.1090/gsm/105
A. S. Lewis, P. A. Parrilo, and M. V. Ramana, The Lax conjecture is true, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2495–2499. MR 2146191, DOI 10.1090/S0002-9939-05-07752-X
Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava, Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem, Ann. of Math. (2) 182 (2015), no. 1, 327–350. MR 3374963, DOI 10.4007/annals.2015.182.1.8
Wim Nuij, A note on hyperbolic polynomials, Math. Scand. 23 (1968), 69–72 (1969). MR 250128, DOI 10.7146/math.scand.a-10898
Adam Parusiński and Armin Rainer, A new proof of Bronshtein’s theorem, J. Hyperbolic Differ. Equ. 12 (2015), no. 4, 671–688. MR 3450058, DOI 10.1142/S0219891615500198
Armin Rainer, Quasianalytic multiparameter perturbation of polynomials and normal matrices, Trans. Amer. Math. Soc. 363 (2011), no. 9, 4945–4977. MR 2806697, DOI 10.1090/S0002-9947-2011-05311-0
Armin Rainer, Perturbation theory for normal operators, Trans. Amer. Math. Soc. 365 (2013), no. 10, 5545–5577. MR 3074382, DOI 10.1090/S0002-9947-2013-05854-0
Franz Rellich, Störungstheorie der Spektralzerlegung, Math. Ann. 113 (1937), no. 1, 600–619 (German). MR 1513109, DOI 10.1007/BF01571652
Franz Rellich, Perturbation theory of eigenvalue problems, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Assisted by J. Berkowitz; With a preface by Jacob T. Schwartz. MR 0240668
R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683, DOI 10.1515/9781400873173
Jakub Duda, Libor Veselý, and Luděk Zajíček, On d.c. functions and mappings, Atti Sem. Mat. Fis. Univ. Modena 51 (2003), no. 1, 111–138. MR 1993883
L. Veselý and L. Zajíček, Delta-convex mappings between Banach spaces and applications, Dissertationes Math. (Rozprawy Mat.) 289 (1989), 52. MR 1016045
Victor Vinnikov, Selfadjoint determinantal representations of real plane curves, Math. Ann. 296 (1993), no. 3, 453–479. MR 1225986, DOI 10.1007/BF01445115