Optimal Sobolev regularity  of roots of polynomials

Parusiński, Adam; Rainer, Armin

doi:10.24033/asens.2376

Optimal Sobolev regularity of roots of polynomials
[Régularité optimale de type Sobolev des racines d'un polynôme]

Parusiński, Adam ; Rainer, Armin

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 5, pp. 1343-1387.

Résumé
Abstract

Nous étudions la régularité des racines d'un polynôme complexe univarié dont les coefficients varient de façon lisse. Nous montrons que tout choix continu de racines d'une $C^{n - 1, 1}$ -courbe de polynômes unitaires de degré $n$ est localement absolument continu avec ses dérivées localement $p$ -intégrables pour tout $1 \leq p < n / (n - 1)$ , uniformément par rapport aux coefficients. Ce résultat est optimal : en général, les dérivées de racines d'une courbe lisse de polynômes unitaires de degré $n$ ne sont pas localement $n / (n - 1)$ -intégrables et la variation des racines peut être localement non bornée si les coefficients sont de classe $C^{n - 1, α}$ pour $α < 1$ . Nous montrons aussi une généralisation des inégalités de Glaeser d'ordre supérieur à la Ghisi et Gobbino. Nous donnons trois applications des résultats principaux : résolution locale d'un système d'équations pseudo-différentielles, un théorème de relèvement pour les applications à valeurs dans l'espace des orbites d'une représentation d'un groupe fini et une condition suffisante pour qu'une fonction multivaluée soit de classe de Sobolev $W^{1, p}$ au sens d'Almgren.

We study the regularity of the roots of complex univariate polynomials whose coefficients depend smoothly on parameters. We show that any continuous choice of a root of a $C^{n - 1, 1}$ -curve of monic polynomials of degree $n$ is locally absolutely continuous with locally $p$ -integrable derivatives for every $1 \leq p < n / (n - 1)$ , uniformly with respect to the coefficients. This result is optimal: in general, the derivatives of the roots of a smooth curve of monic polynomials of degree $n$ are not locally $n / (n - 1)$ -integrable, and the roots may have locally unbounded variation if the coefficients are only of class $C^{n - 1, α}$ for $α < 1$ . We also prove a generalization of Ghisi and Gobbino's higher order Glaeser inequalities. We give three applications of the main results: local solvability of a system of pseudo-differential equations, a lifting theorem for mappings into orbit spaces of finite group representations, and a sufficient condition for multi-valued functions to be of Sobolev class $W^{1, p}$ in the sense of Almgren.

Publié le : 2018-11-12

DOI : 10.24033/asens.2376

Classification : 26C10, 26A46, 26D10, 30C15, 46E35
Keywords: Perturbation of complex polynomials, absolute continuity of roots, optimal regularity of the roots among Sobolev spaces $W^{1,p}$, higher order Glaeser inequalities
Mot clés : Perturbations des polynômes complexes, continuité absolue des racines, régularité optimale des racines dans les espaces de Sobolev $W^{1,p}$, inégalités de Glaeser d'ordre supérieur

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     author = {Parusi\'nski, Adam and Rainer, Armin},
     title = {Optimal {Sobolev} regularity  of roots of polynomials},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1343--1387},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
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     doi = {10.24033/asens.2376},
     mrnumber = {3942042},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2376/}
}

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Parusiński, Adam; Rainer, Armin. Optimal Sobolev regularity  of roots of polynomials. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 5, pp. 1343-1387. doi : 10.24033/asens.2376. http://archive.numdam.org/articles/10.24033/asens.2376/

Bibliographie
Cité par

Alekseevsky, D.; Kriegl, A.; Losik, M.; Michor, P. W. Lifting smooth curves over invariants for representations of compact Lie groups, Transform. Groups, Volume 5 (2000), pp. 103-110 (ISSN: 1083-4362) | DOI | MR | Zbl

Alekseevsky, D.; Kriegl, A.; Michor, P. W.; Losik, M. Choosing roots of polynomials smoothly, Israel J. Math., Volume 105 (1998), pp. 203-233 (ISSN: 0021-2172) | DOI | MR | Zbl

Almgren, F. J. J., World Scientific Monograph Series in Mathematics, 1, World Scientific Publishing Co., Inc., River Edge, NJ, 2000, 955 pages (ISBN: 981-02-4108-9) | MR | Zbl

Bony, J.-M.; Broglia, F.; Colombini, F.; Pernazza, L. Nonnegative functions as squares or sums of squares, J. Funct. Anal., Volume 232 (2006), pp. 137-147 (ISSN: 0022-1236) | DOI | MR | Zbl

Beck, T.; Becker-Kahn, S.; Hanin, B. Nodal sets of smooth functions with finite vanishing order and $p$ -sweepouts, Calc. Var., Volume 57 (2018), 140 pages | DOI | MR

Bony, J.-M.; Colombini, F.; Pernazza, L., Phase space analysis of partial differential equations (Progr. Nonlinear Differential Equations Appl.), Volume 69, Birkha̋user Boston, Boston, MA, 2006, pp. 45-53 | DOI | MR | Zbl

Bony, J.-M.; Colombini, F.; Pernazza, L. On square roots of class $C^{m}$ of nonnegative functions of one variable, Ann. Sc. Norm. Super. Pisa Cl. Sci., Volume 9 (2010), pp. 635-644 (ISSN: 0391-173X) | Numdam | MR | Zbl

Bierstone, E.; Milman, P. D. Arc-analytic functions, Invent. math., Volume 101 (1990), pp. 411-424 (ISSN: 0020-9910) | DOI | MR | Zbl

Bronšteĭn, M. D. Smoothness of roots of polynomials depending on parameters, Sibirsk. Mat. Zh., Volume 20 (1979), p. 493-501, 690 ; English translation: Siberian Math. J. 20 (1980), 347–352 (ISSN: 0037-4474) | MR | Zbl

Chaumat, J.; Chollet, A.-M. Division par un polynôme hyperbolique, Canad. J. Math., Volume 56 (2004), pp. 1121-1144 (ISSN: 0008-414X) | DOI | MR | Zbl

Colombini, F.; Jannelli, E.; Spagnolo, S. Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coefficients depending on time, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 10 (1983), pp. 291-312 (ISSN: 0391-173X) | Numdam | MR | Zbl

Colombini, F.; Lerner, N. Une procedure de Calderón-Zygmund pour le problème de la racine $k$ -ième, Ann. Mat. Pura Appl., Volume 182 (2003), pp. 231-246 (ISSN: 0373-3114) | DOI | MR | Zbl

Colombini, F.; Orrù, N.; Pernazza, L. On the regularity of the roots of hyperbolic polynomials, Israel J. Math., Volume 191 (2012), pp. 923-944 (ISSN: 0021-2172) | DOI | MR | Zbl

De Lellis, C.; Spadaro, E. N. $Q$ -valued functions revisited, Mem. Amer. Math. Soc., Volume 211 (2011) (ISBN: 978-0-8218-4914-9, ISSN: 0065-9266) | DOI | MR | Zbl

D'Ancona, P.; Spagnolo, S. Quasi-symmetrization of hyperbolic systems and propagation of the analytic regularity, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., Volume 1 (1998), pp. 169-185 (ISSN: 0392-4041) | MR | Zbl

Ghisi, M.; Gobbino, M. Higher order Glaeser inequalities and optimal regularity of roots of real functions, Ann. Sc. Norm. Super. Pisa Cl. Sci., Volume 12 (2013), pp. 1001-1021 (ISSN: 0391-173X) | Numdam | MR | Zbl

Glaeser, G. Racine carrée d'une fonction différentiable, Ann. Inst. Fourier (Grenoble), Volume 13 (1963), pp. 203-210 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Grafakos, L., Graduate Texts in Math., 249, Springer, New York, 2008, 489 pages (ISBN: 978-0-387-09431-1) | MR | Zbl

Jannelli, E. On the symmetrization of the principal symbol of hyperbolic equations, Comm. Partial Differential Equations, Volume 14 (1989), pp. 1617-1634 (ISSN: 0360-5302) | DOI | MR

Kato, T., Grundl. math. Wiss., 132, Springer, Berlin-New York, 1976 | MR | Zbl

Kriegl, A.; Losik, M.; Michor, P. W. Choosing roots of polynomials smoothly. II, Israel J. Math., Volume 139 (2004), pp. 183-188 (ISSN: 0021-2172) | DOI | MR | Zbl

Kriegl, A.; Losik, M.; Michor, P. W.; Rainer, A. Lifting smooth curves over invariants for representations of compact Lie groups. II, J. Lie Theory, Volume 15 (2005), pp. 227-234 (ISSN: 0949-5932) | MR | Zbl

Kriegl, A.; Losik, M.; Michor, P. W.; Rainer, A. Lifting smooth curves over invariants for representations of compact Lie groups. III, J. Lie Theory, Volume 16 (2006), pp. 579-600 (ISSN: 0949-5932) | MR | Zbl

Kriegl, A.; Losik, M.; Michor, P. W.; Rainer, A. Lifting mappings over invariants of finite groups, Acta Math. Univ. Comenian. (N.S.), Volume 77 (2008), pp. 93-122 (ISSN: 0862-9544) | MR | Zbl

Kriegl, A.; Losik, M.; Michor, P. W.; Rainer, A. Addendum to: “Lifting smooth curves over invariants for representations of compact Lie groups, III” J. Lie Theory 16 (2006), No. 3, 579–600. [MR2248147], J. Lie Theory, Volume 22 (2012), pp. 245-249 (ISSN: 0949-5932) | MR | Zbl

Losik, M.; Michor, P. W.; Rainer, A. A generalization of Puiseux's theorem and lifting curves over invariants, Rev. Mat. Complut., Volume 25 (2012), pp. 139-155 (ISSN: 1139-1138) | DOI | MR | Zbl

Malgrange, B., Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Tata Institute of Fundamental Research, Bombay; Oxford Univ. Press, London, 1967, 106 pages | MR | Zbl

Mandai, T. Smoothness of roots of hyperbolic polynomials with respect to one-dimensional parameter, Bull. Fac. Gen. Ed. Gifu Univ., Volume 21 (1985), pp. 115-118 (ISSN: 0286-3251) | MR

Marcus, M.; Mizel, V. J. Complete characterization of functions which act, via superposition, on Sobolev spaces, Trans. Amer. Math. Soc., Volume 251 (1979), pp. 187-218 (ISSN: 0002-9947) | DOI | MR | Zbl

Parusiński, A.; Rainer, A. A new proof of Bronshtein's theorem, J. Hyperbolic Differ. Equ., Volume 12 (2015), pp. 671-688 (ISSN: 0219-8916) | DOI | MR

Parusiński, A.; Rainer, A. Lifting differentiable curves from orbit spaces, Transform. Groups, Volume 21 (2016), pp. 153-179 (ISSN: 1083-4362) | DOI | MR

Parusiński, A.; Rainer, A. Regularity of roots of polynomials, Ann. Sc. Norm. Super. Pisa Cl. Sci., Volume 16 (2016), pp. 481-517 (ISSN: 0391-173X) | MR

Rainer, A. Perturbation of complex polynomials and normal operators, Math. Nachr., Volume 282 (2009), pp. 1623-1636 (ISSN: 0025-584X) | DOI | MR | Zbl

Rainer, A. Quasianalytic multiparameter perturbation of polynomials and normal matrices, Trans. Amer. Math. Soc., Volume 363 (2011), pp. 4945-4977 (ISSN: 0002-9947) | DOI | MR | Zbl

Rainer, A. Smooth roots of hyperbolic polynomials with definable coefficients, Israel J. Math., Volume 184 (2011), pp. 157-182 (ISSN: 0021-2172) | DOI | MR | Zbl

Rainer, A. Lifting quasianalytic mappings over invariants, Canad. J. Math., Volume 64 (2012), pp. 409-428 (ISSN: 0008-414X) | DOI | MR | Zbl

Rainer, A. Perturbation theory for normal operators, Trans. Amer. Math. Soc., Volume 365 (2013), pp. 5545-5577 (ISSN: 0002-9947) | DOI | MR | Zbl

Rainer, A. Differentiable roots, eigenvalues, and eigenvectors, Israel J. Math., Volume 201 (2014), pp. 99-122 (ISSN: 0021-2172) | DOI | MR | Zbl

Rahman, Q. I.; Schmeisser, G., London Mathematical Society Monographs. New Series, 26, The Clarendon Press, Oxford Univ. Press, Oxford, 2002, 742 pages (ISBN: 0-19-853493-0) | MR | Zbl

Spagnolo, S. Local and semi-global solvability for systems of non-principal type, Comm. Partial Differential Equations, Volume 25 (2000), pp. 1115-1141 (ISSN: 0360-5302) | DOI | MR | Zbl

Spagnolo, S. On the absolute continuity of the roots of some algebraic equations, Ann. Univ. Ferrara Sez. VII (N.S.), Volume 45 (1999), pp. 327-337 (ISSN: 0430-3202) | DOI | MR | Zbl

Stein, E. M., Princeton Mathematical Series, No. 30, Princeton Univ. Press, Princeton, N.J., 1970, 290 pages | MR | Zbl

Tarama, S. On the lemma of Colombini, Jannelli and Spagnolo, Memoirs of the Faculty of Engineering, Osaka City University, Volume 41 (2000), pp. 111-115

Tarama, S. Note on the Bronshtein theorem concerning hyperbolic polynomials, Sci. Math. Jpn., Volume 63 (2006), pp. 247-285 (ISSN: 1346-0862) | MR | Zbl

Wakabayashi, S. Remarks on hyperbolic polynomials, Tsukuba J. Math., Volume 10 (1986), pp. 17-28 (ISSN: 0387-4982) | DOI | MR | Zbl

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