  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 $P {\o}\pi_k$ 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 $SBV_{loc}$-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 $1 < p \le \infty$. 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 $SBV_{loc}$.

Classical Analysis and ODEs;Algebraic Geometry