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Moscow Mathematical Journal, 2005, Volume 5, Number 4, Pages 781–828
DOI: https://doi.org/10.17323/1609-4514-2005-5-4-781-828 (Mi mmj223) 		 
This article is cited in 4 scientific papers (total in 4 papers)

Upper bounds of topology of complex polynomials in two variables

A. A. Glutsyukab

a Independent University of Moscow
b CNRS — Unit of Mathematics, Pure and Applied
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DOI: https://doi.org/10.17323/1609-4514-2005-5-4-781-828
Abstract: It is well known that the roots and critical points of a complex polynomial in one variable admit an explicit upper bound in terms of the highest coefficient and the maximal modulus of critical values. In the present paper, we prove similar bounds for generic complex polynomials in two variables. In particular, we give an upper bound of the radii of bidisks that contain all the nontrivial topology of level curves. These results were used in studying the restricted version of Hilbert's 16th problem in the joint paper of the author with Yu. S. Ilyashenko.
Key words and phrases: Complex polynomial in two variables, level curve, bidisk containing the topology of level curve, critical point, critical value, vanishing cycle, abelian integral, period determinant.
Received: July 17, 2005
Bibliographic databases:
MSC: 14D05, 30C15, 30C10
Language: English
Citation: A. A. Glutsyuk, “Upper bounds of topology of complex polynomials in two variables”, Mosc. Math. J., 5:4 (2005), 781–828
Citation in format AMSBIB
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\by A.~A.~Glutsyuk
\paper Upper bounds of topology of complex polynomials in two variables
\jour Mosc. Math.~J.
\yr 2005
\vol 5
\issue 4
\pages 781--828
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\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2266460}
\zmath{https://zbmath.org/?q=an:1186.14013}
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