Die u:cris Detailansicht:
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