  We show that a normal matrix $A$ with coefficient in $\mathbb C[[X]]$,
$X=(X_1, \ldots, X_n)$, can be diagonalized, provided the discriminant
$\Delta_A $ of its characteristic polynomial is a monomial times a unit. The
proof is an adaptation of the algorithm of proof of Abhyankar-Jung Theorem. As
a corollary we obtain the singular value decomposition for an arbitrary matrix
$A$ with coefficient in $\mathbb C[[X]]$ under a similar assumption on
$\Delta_{AA^*} $ and $\Delta_{A^*A} $.
  We also show real versions of these results, i.e. for coefficients in
$\mathbb R[[X]]$, and deduce several results on multiparameter perturbation
theory for normal matrices with real analytic, quasi-analytic, or Nash
coefficients.
