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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quasianalytic multiparameter perturbation of polynomials and normal matrices


Author: Armin Rainer
Journal: Trans. Amer. Math. Soc. 363 (2011), 4945-4977
MSC (2010): Primary 26C10, 26E10, 30C15, 32B20, 47A55, 47A56
Published electronically: April 14, 2011
MathSciNet review: 2806697
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Abstract: We study the regularity of the roots of multiparameter families of complex univariate monic polynomials $ P(x)(z) = z^n + \sum_{j=1}^n (-1)^j a_j(x) z^{n-j} $ with fixed degree $ n$ whose coefficients belong to a certain subring $ \mathcal{C}$ of $ C^\infty$-functions. We require that $ \mathcal{C}$ includes polynomials but excludes flat functions (quasianalyticity) and is closed under composition, derivation, division by a coordinate, and taking the inverse. Examples are quasianalytic Denjoy-Carleman classes, in particular, the class of real analytic functions $ C^\omega$.

We show that there exists a locally finite covering $ \{\pi_k\}$ of the parameter space, where each $ \pi_k$ is a composite of finitely many $ \mathcal{C}$-mappings, each of which is either a local blow-up with smooth center or a local power substitution (in coordinates given by $ x \mapsto (\pm x_1^{\gamma_1},\ldots,\pm x_q^{\gamma_q})$, $ \gamma_i \in \mathbb{N}_{>0}$), such that, for each $ k$, the family of polynomials $ P \circ \pi_k$ admits a $ \mathcal{C}$-parameterization of its roots. If $ P$ is hyperbolic (all roots real), then local blow-ups suffice.

Using this desingularization result, we prove that the roots of $ P$ can be parameterized by $ SBV_{\operatorname{loc}}$-functions whose classical gradients exist almost everywhere and belong to $ L^1_{\operatorname{loc}}$. In general the roots cannot have gradients in $ L^p_{\operatorname{loc}}$ for any $ 1 < p \le \infty$. Neither can the roots be in $ W_{\operatorname{loc}}^{1,1}$ or $ VMO$.

We obtain the same regularity properties for the eigenvalues and the eigenvectors of $ \mathcal{C}$-families of normal matrices. A further consequence is that every continuous subanalytic function belongs to $ SBV_{\operatorname{loc}}$.


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Additional Information

Armin Rainer
Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria
Email: armin.rainer@univie.ac.at

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05311-0
PII: S 0002-9947(2011)05311-0
Keywords: Quasianalytic, Denjoy–Carleman class, multiparameter perturbation theory, smooth roots of polynomials, desingularization, bounded variation, subanalytic
Received by editor(s): January 28, 2010
Published electronically: April 14, 2011
Additional Notes: The author was supported by the Austrian Science Fund (FWF), Grant J2771
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.