Quasianalytic multiparameter perturbation of polynomials and normal matrices

Abstract

We study the regularity of the roots of multiparameter families of complex univariate monic polynomials with fixed degree n whose coefficients belong to a certain subring C of C∞- functions. We require that 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 Cω. We show that there exists a locally finite covering {πk} of the parameter space, where each πk is a composite of finitely many C-mappings, each of which is either a local blow-up with smooth center or a local power substitution (in coordinates given by x m (±x 1 γ 1 ,...,±x q γ q ), γi ∈ ℕ > 0), such that, for each k, the family of polynomials P pk admits a 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 loc -functions whose classical gradients exist almost everywhere and belong to L 1 loc . In general the roots cannot have gradients in L p loc for any 1 < p=8. Neither can the roots be in W 1,1 loc or VMO. We obtain the same regularity properties for the eigenvalues and the eigenvectors of C-families of normal matrices. A further consequence is that every continuous subanalytic function belongs to SBV loc . © 2011 American Mathematical Society.