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 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 $SBV_{\operatorname {loc}}$.