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Hyperbolic polynomials and multiparameter real-analytic perturbation theory. (English) Zbl 1140.15006

Let \(P\) be the polynomial defined by \[ P(x,z)=z^d+\sum_{i=1}^da_i(x)z^{d-i}\; , \] where each \(a_i\) is a real-analytic function on an open interval \(I\subset\mathbb{R}\). F. Rellich [Math. Ann. 113, 600–619 (1936; Zbl 0016.06201)] proved that if, for any \(x\in I\), the polynomial \(x\mapsto P(x,z)\) is hyperbolic, i.e., it has only real roots, then the roots of \(P\) can be chosen analytically.
In this paper, the authors provide two multiparameter generalizations of Rellich’s theory. For the first generalization, it is proved that the roots of \(P\) can be chosen locally in a Lipschitz way. For the second generalization, after a suitable blowup of the space of parameters, they write locally the roots of hyperbolic polynomials as analytic functions of the parameters.
In the second half of the paper, another result by F. Rellich is generalized [Perturbation theory of eigenvalue problems. Notes on Mathematics and its Applications. New York-London-Paris: Gordon and Breach Science Publishers (1969; Zbl 0181.42002)] stating that a 1-parameter analytic family of symmetric matrices admits a uniform diagonalization. In the end, analytic families of antisymmetric matrices, depending on several parameters, are also considered.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
32B20 Semi-analytic sets, subanalytic sets, and generalizations
14P20 Nash functions and manifolds
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