Abstract
Let be a polynomial, where are real-analytic functions in an open subset of . If, for any , the polynomial has only real roots, then we can write those roots as locally Lipschitz functions of . Moreover, there exists a modification (a locally finite composition of blowups with smooth centers) such that the roots of the corresponding polynomial , can be written locally as analytic functions of . Let , be an analytic family of symmetric matrices, where is open in . Then there exists a modification such that the corresponding family 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 -parameter families. Similarly, for an analytic family , of antisymmetric matrices, there exists a modification such that we can find locally a basis of proper subspaces in an analytic way
Citation
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
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