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 ${\mathbb{R}}^n$. 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 blowups 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 ${\mathbb{R}}^n$. Then there exists a modification $\sigma :W\to U$ such that the corresponding family $\tilde{A}(w)=A(\sigma (w))$ can be locally diagonalized analytically (i.e., we can choose locally a basis of eigenvectors in an analytic way). This generalizes Rellich's well-known theorem (see [32]) from 1937 for $1$-parameter families. Similarly, for an analytic family $A(x),x\in U$, of antisymmetric matrices, there exists a modification $\sigma$ such that we can find locally a basis of proper subspaces in an analytic way