Distribution of critical points of polynomials
Author:
Vilmos Totik
Journal:
Trans. Amer. Math. Soc. 372 (2019), 2407-2428
MSC (2010):
Primary 26C10, 31A15
DOI:
https://doi.org/10.1090/tran/7667
Published electronically:
May 23, 2019
MathSciNet review:
3988581
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Abstract | References | Similar Articles | Additional Information
Abstract: The problem of the distribution of the critical points of polynomials in terms of the distribution $\mu$ of the zeros is considered. It is shown that away from the inner boundary of the (compact) support $S$ of $\mu$ the two distributions are the same. This is the case, in particular, if $S$ has connected complement. Examples are given showing that the two distributions may not be the same everywhere if the inner boundary has positive $\mu$-measure, but it is also shown that such examples are rare and very unstable.
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Additional Information
Vilmos Totik
Affiliation:
MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Szeged, Aradi v. tere 1, 6720, Hungary; and Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Avenue CMC342, Tampa, Florida 33620
Email:
totik@mail.usf.edu
Keywords:
Distribution of critical points,
polynomials,
Cauchy-transform,
potential theory
Received by editor(s):
August 28, 2017
Received by editor(s) in revised form:
April 24, 2018
Published electronically:
May 23, 2019
Additional Notes:
The author was supported by NSF grant DMS 1564541
Article copyright:
© Copyright 2019
American Mathematical Society