Quasianalytic multiparameter perturbation of polynomials and normal matrices

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 polynomial 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 $x\mapsto (\pm x_1^{{\gamma }_1},...,\pm x_q^{{\gamma }_q})$, ${\gamma }_i\in {\mathbb{N}}_{>0}$), such that, for each $k$, the family of polynomials $Pø{\pi }_k$ 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 $SB{V}_{loc}$-functions whose classical gradients exist almost everywhere and belong to $L_{loc}^1$. In general the roots cannot have gradients in $L_{loc}^p$ for any $1<p\le \infty$. Neither can the roots be in $W_{loc}^{1,1}$ 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 $SB{V}_{loc}$.

33 pages, 1 figure, minor corrections, to appear in Trans. Amer. Math. Soc

Cited by in corpus (4)

• Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators
• Differentiable roots, eigenvalues, and eigenvectors
• Re-parameterizing and reducing families of normal operators
• Selections of bounded variation for roots of smooth polynomials