Hyperbolic polynomials and multiparameter real analytic perturbation theory

Details Hyperbolic polynomials and multiparameter real analytic perturbation theory Hyperbolic polynomials and multiparameter real analytic perturbation theory

2006-02-23

arxiv.org/abs/math/0602538v1

Mathematics

General Mathematics

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Hyperbolic polynomials and multiparameter real analytic perturbation theory does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Hyperbolic polynomials and multiparameter real analytic perturbation theory, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hyperbolic polynomials and multiparameter real analytic perturbation theory will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-711713