Hyperbolic polynomials and multiparameter real analytic perturbation theory
| dc.creator | Kurdyka, Krzysztof | |
| dc.creator | Paunescu, Laurentiu | |
| dc.date | 2006-02-23 | |
| dc.date.accessioned | 2026-07-07T07:03:44Z | |
| dc.date.available | 2026-07-07T07:03:44Z | |
| dc.description | 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) $σ: W \to U$ such that the roots of the corresponding polynomial $\tilde P(w,z) =P(σ(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 $σ: W \to U$, such the corresponding family $\tilde A(w) =A(σ(w))$ 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 $σ$ such that we can find locally a basis of proper subspaces in an analytic way. | |
| dc.identifier | https://arxiv.org/abs/math/0602538 | |
| dc.identifier | http://arxiv.org/abs/math/0602538 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/109088 | |
| dc.subject | General Mathematics | |
| dc.subject | 15A18;32B20;14B20 | |
| dc.title | Hyperbolic polynomials and multiparameter real analytic perturbation theory | |
| dc.type | text |