Duke Mathematical Journal

Hyperbolic polynomials and multiparameter real-analytic perturbation theory

Krzysztof Kurdyka and Laurentiu Paunescu

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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

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Duke Math. J. Volume 141, Number 1 (2008), 123-149.

First available in Project Euclid: 4 December 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A18: Eigenvalues, singular values, and eigenvectors
Secondary: 32B20: Semi-analytic sets and subanalytic sets [See also 14P15] 14P20: Nash functions and manifolds [See also 32C07, 58A07]


Kurdyka, Krzysztof; Paunescu, Laurentiu. Hyperbolic polynomials and multiparameter real-analytic perturbation theory. Duke Math. J. 141 (2008), no. 1, 123--149. doi:10.1215/S0012-7094-08-14113-4. http://projecteuclid.org/euclid.dmj/1196794292.

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