Quasianalytic perturbation of multi-parameter hyperbolic polynomials and symmetric matrices

This paper investigates hyperbolic polynomials with quasianalytic coefficients. Our main purpose is to prove factorization theorems for such polynomials, and next to generalize the results of K. Kurdyka and L. Paunescu about perturbation of analytic families of symmetric matrices to the quasianalytic settings. Generally, the perturbation problem concerns the issue whether, given a family of monic polynomials with coefficients from a certain class of functions, one can represent its roots as function of this class. Hyperbolic polynomials with analytic coefficients in one variable were studied by Rellich [24, 25], which was linked with his investigation into the behaviour of eigenvalues of symmetric matrices under one-parameter analytic perturbation. This oneparameter theory, initiated by Rellich, culminated in the work of Kato [8]. One-parameter families of hyperbolic polynomials were contemporarily studied in [2, 10], as well. Recently, Kurdyka–Paunescu [12] developed multiparameter analytic perturbation theory. Our purpose is to carry over this multi-parameter theory to the quasianalytic settings. This task is also the subject of a contemporaneous paper [23]. The main purpose of this paper is to establish certain splitting theorems for quasiordinary hyperbolic polynomials with quasianalytic coefficients. Our 2010 MSC: 14P15, 32B20, 15A18, 26E10.