# Hyperbolic polynomials and multiparameter real analytic perturbation
theory

Research paper by **Krzysztof Kurdyka, Laurentiu Paunescu**

Indexed on: **23 Feb '06**Published on: **23 Feb '06**Published in: **Mathematics - General Mathematics**

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#### 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.