Hyperbolic polynomials and multiparameter real analytic perturbation theory

Abstract

Let $P(x,z)= z^d +\sum_{i=1}^{d}a_i(x)z^{d-i}$ be a polynomial, where $a_i$ are real analytic functions in an open subset $U$ of $\R^n$. If for any $x \in U$ the polynomial $z\mapsto P(x,z)$ has only real roots, then we can write those roots as locally lipschitz functions of $x$. Moreover, there exists a modification (a locally finite composition of blowing-ups with smooth centers) $\sigma : W \to U$ such that the roots of the corresponding polynomial $\tilde P(w,z) =P(\sigma (w),z), w\in W $, can be written locally as analytic functions of $w$. Let $A(x), x\in U$ be an analytic family of symmetric matrices, where $U$ is open in $\R^n$. Then there exists a modification $\sigma : W \to U$, such the corresponding family $\tilde A(w) =A(\sigma(w))$ can be locally diagonalized analytically (i.e. we can choose locally eigenvectors in an analytic way). This generalizes the Rellich's well known theorem (1937) for one-parameter families. Similarly for an analytic family $A(x), x\in U$ of antisymmetric matrices there exits a modification $\sigma$ such that we can find locally a basis of proper subspaces in an analytic way.