Selections of bounded variation for roots of smooth polynomials

Autor(en)

Adam Parusinski, Armin Rainer

Abstrakt

We prove that the roots of a smooth monic polynomial with complex-valued coefficients defined on a bounded Lipschitz domain Ω in Rm admit a parameterization by functions of bounded variation uniformly with respect to the coefficients. This result is best possible in the sense that discontinuities of the roots are in general unavoidable due to monodromy. We show that the discontinuity set can be chosen to be a finite union of smooth hypersurfaces. On its complement the parameterization of the roots is of optimal Sobolev class W1,p for all 1≤p<nn−1, where n is the degree of the polynomial. All discontinuities are jump discontinuities. For all this we require the coefficients to be of class Ck−1,1(Ω¯¯¯¯), where k is a positive integer depending only on n and m. The order of differentiability k is not optimal. However, in the case of radicals, i.e., for the solutions of the equation Zr=f, where f is a complex-valued function and r∈R>0, we obtain optimal uniform bounds.

Organisation(en)

Institut für Mathematik

Externe Organisation(en)

Pädagogische Hochschule Niederösterreich, Université Côte d'Azur

Journal

Selecta Mathematica

Band

26

Anzahl der Seiten

40

ISSN

1022-1824

DOI

https://doi.org/10.1007/s00029-020-0538-z

Publikationsdatum

2020

Peer-reviewed

Ja

ÖFOS 2012

101002 Analysis

Schlagwörter

ASJC Scopus Sachgebiete

Allgemeine Physik und Astronomie, Allgemeine Mathematik

Link zum Portal

https://ucrisportal.univie.ac.at/de/publications/b027c6f9-ffb3-4fcf-b2f0-579eeeae68e5