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15 January 2008 Hyperbolic polynomials and multiparameter real-analytic perturbation theory
Krzysztof Kurdyka, Laurentiu Paunescu
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Duke Math. J. 141(1): 123-149 (15 January 2008). DOI: 10.1215/S0012-7094-08-14113-4

Abstract

Let P(x,z)=zd+i=1dai(x)zd-i be a polynomial, where ai are real-analytic functions in an open subset U of Rn. If, for any xU, the polynomial zP(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) σ:WU such that the roots of the corresponding polynomial P~(w,z)=P(σ(w),z),wW, can be written locally as analytic functions of w. Let A(x),xU, be an analytic family of symmetric matrices, where U is open in Rn. Then there exists a modification σ:WU such that the corresponding family A~(w)=A(σ(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),xU, of antisymmetric matrices, there exists a modification σ such that we can find locally a basis of proper subspaces in an analytic way

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Krzysztof Kurdyka. Laurentiu Paunescu. "Hyperbolic polynomials and multiparameter real-analytic perturbation theory." Duke Math. J. 141 (1) 123 - 149, 15 January 2008. https://doi.org/10.1215/S0012-7094-08-14113-4

Information

Published: 15 January 2008
First available in Project Euclid: 4 December 2007

zbMATH: 1140.15006
MathSciNet: MR2372149
Digital Object Identifier: 10.1215/S0012-7094-08-14113-4

Subjects:
Primary: 15A18‎
Secondary: 14P20, 32B20

Rights: Copyright © 2008 Duke University Press

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Vol.141 • No. 1 • 15 January 2008
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