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Transactions of the American Mathematical Society

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Distribution of critical points of polynomials


Author: Vilmos Totik
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 26C10, 31A15
DOI: https://doi.org/10.1090/tran/7667
Published electronically: May 23, 2019
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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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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

DOI: https://doi.org/10.1090/tran/7667
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