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 Mosc. Math. J., 2011, Volume 11, Number 3, Pages 617–625 (Mi mmj436)

Topological complexity and Schwarz genus of general real polynomial equation

V. A. Vassilievab

a Steklov Mathematical Institute, Moscow, Russia
b Mathematics Department, Higher School of Economics, Moscow, Russia

Abstract: We prove that the minimal number of branchings of arithmetic algorithms of approximate solution of the general real polynomial equation $x^d+a_1x^{d-1}+…+a_{d-1}x+a_d=0$ of odd degree $d$ grows to infinity at least as $\log_2d$. The same estimate is true for the $\varepsilon$-genus of the real algebraic function associated with this equation, i.e. for the minimal number of open sets covering the space $\mathbb R^d$ of such polynomials in such a way that on any of these sets there exists a continuous function whose value at any point $(a_1,…,a_d)$ is approximately (up to some sufficiently small $\varepsilon>0$) equal to one of real roots of the corresponding equation.

Key words and phrases: complexity, cross-section, Schwarz genus, ramified covering, 13th Hilbert problem, real polynomial.

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Bibliographic databases:
MSC: Primary 55R80, 12Y05; Secondary 55S40, 68W30
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Citation: V. A. Vassiliev, “Topological complexity and Schwarz genus of general real polynomial equation”, Mosc. Math. J., 11:3 (2011), 617–625

Citation in format AMSBIB
\Bibitem{Vas11} \by V.~A.~Vassiliev \paper Topological complexity and Schwarz genus of general real polynomial equation \jour Mosc. Math.~J. \yr 2011 \vol 11 \issue 3 \pages 617--625 \mathnet{http://mi.mathnet.ru/mmj436} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2894434} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000300365900012} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. A. Vassiliev, “On topological invariants of real algebraic functions”, Funct. Anal. Appl., 45:3 (2011), 163–172