Perturbation of complex polynomials and normal operators

Abstract

Key words Regular roots of polynomials, absolute continuity, perturbation of normal operators MSC (2000) Primary: 26C10, 30C15, 47A55, 47A56We study the regularity of the roots of complex monic polynomials P (t) of fixed degree depending smoothly on a real parameter t. We prove that each continuous parameterization of the roots of a generic C ∞ curve P (t) (which always exists) is locally absolutely continuous. Generic means that no two of the continuously chosen roots meet of infinite order of flatness. Simple examples show that one cannot expect a better regularity than absolute continuity. This result will follow from the proposition that for any t0 there exists a positive integer N such that t → P t0 ± (t − t0) N admits smooth parameterizations of its roots near t0. We show that C n curves P (t) (where n = deg P ) admit differentiable roots if and only if the order of contact of the roots is ≥ 1. We give applications to the perturbation theory of normal matrices and unbounded normal operators with compact resolvents and common domain of definition: The eigenvalues and eigenvectors of a generic C ∞ curve of such operators can be arranged locally in an absolutely continuous way.

Smooth roots of hyperbolic polynomials with definable coefficients

“…For definable continuous curves of complex polynomials, we show that any continuous choice of roots is actually locally absolutely continuous (not better!). This extends results in [10].…”

“…(IIa) If m 0 (a k ) < ∞ for some 2 ≤ k ≤ n, there exist N, r ∈ N >0 such that (t → P (±t N )) (r) (the reduced curve of polynomials defined in (4.12.1) associated to t → P (±t N )) has distinct roots at t = 0 (see [10]). By the splitting lemma 4.2 and the induction hypothesis, we are done.…”

Higher order Glaeser inequalities and optimal regularity of roots of real functions

“…Spagnolo's result was in some sense extended by A. Rainer [11], who proved the absolute continuity of roots of monic polynomials of arbitrary degree with coefficients of class C 1 , but with the assumption that one can continuously arrange roots in such a way that no two of them meet with an infinite order of flatness. In the case of the polynomial P(y) := y k g(x), namely in the case of k-th roots, this implies that g has a finite number of zeroes.…”

Re-parameterizing and reducing families of normal operators

“…They deduce the analog result for real analytic families of anti-symmetric matrices. Around the same time, Rainer have started a series of papers about (mostly) roots of regular monic complex polynomials [Rai1,Rai2,Rai3,Rai4], and finds analogues of Kurdyka & Paunescu result for multi-variate quasi-analytic families of monic complex polynomials, which when applied to a quasi-analytic family of complex normal matrices, provides, up to a quasi-analytic re-parameterization of the family, the possibility to choose locally quasi-analytically the eigen-values as well as a local unitary frame of eigen-vectors.…”