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},
     number = {5},
     year = {2018},
     doi = {10.24033/asens.2376},
     mrnumber = {3942042},
     language = {en},
     url = {http://www.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://www.numdam.org/articles/10.24033/asens.2376/

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