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 Keldysh Institute preprints, 2012, 077, 25 pp. (Mi ipmp95)

Geometry of Hermite-Padé approximants for system of functions $\{f,f^2\}$ with three branch points

A. I. Aptekarev, D. N. Tulyakov

Abstract: In the problem on asymptotic of Hermite-Padé approximants for two analytic functions with branch points an algebraic function of the third order appears as the Cauchy transform of the limiting measure of poles distributions of the approximants. In general stuation this statement is known as the Nuttall’s conjecture. Our goal is, assuming that this conjecture holds true to describe the algebraic functions for the case when approximated two functions have common three branch points. In this preprint we discuss statement of the problem, general approaches to its solutions, and we carry out analysis of the appearing algebraic functions of genus zero. We plan to consider the cases corresponding to the algebraic functions of higher genus in the future paper.

Keywords: Algebraic functions, Riemann surfaces, Hermite-Pade approximants

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Citation: A. I. Aptekarev, D. N. Tulyakov, “Geometry of Hermite-Padé approximants for system of functions $\{f,f^2\}$ with three branch points”, Keldysh Institute preprints, 2012, 077, 25 pp.

Citation in format AMSBIB
\Bibitem{AptTul12} \by A.~I.~Aptekarev, D.~N.~Tulyakov \paper Geometry of Hermite-Pad{\'e} approximants for system of functions $\{f,f^2\}$ with three branch points \jour Keldysh Institute preprints \yr 2012 \papernumber 077 \totalpages 25 \mathnet{http://mi.mathnet.ru/ipmp95} 

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This publication is cited in the following articles:
1. R. K. Kovacheva, S. P. Suetin, “Distribution of zeros of the Hermite–Padé polynomials for a system of three functions, and the Nuttall condenser”, Proc. Steklov Inst. Math., 284 (2014), 168–191
2. A. I. Aptekarev, D. N. Tulyakov, “Abelev integral Nattolla na rimanovoi poverkhnosti kubicheskogo kornya mnogochlena 3-i stepeni”, Preprinty IPM im. M. V. Keldysha, 2014, 015, 25 pp.
3. S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951
4. A. I. Aptekarev, D. N. Tulyakov, “Nuttall's Abelian integral on the Riemann surface of the cube root of a polynomial of degree 3”, Izv. Math., 80:6 (2016), 997–1034
5. V. G. Lysov, “Silnaya asimptotika approksimatsii Ermita–Pade dlya sistemy Nikishina s vesami Yakobi”, Preprinty IPM im. M. V. Keldysha, 2017, 085, 35 pp.
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