Hyperelliptic uniformization of algebraic curves of the third order

An algebraic function of the third order plays an important role in the problem of asymptotics of Hermite–Padé approximants for two analytic functions with branch points. This algebraic function appears as the Cauchy transform of the limiting measure of the asymptotic distribution of the poles of th...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2015-08, Vol.284, p.38-49
Main Authors: Aptekarev, A.I., Toulyakov, D.N., Van Assche, W.
Format: Article
Language:English
Subjects:
Algebra
Algebraic functions
Approximants
Approximation
Asymptotic properties
Constraining
Hermite–Padé rational approximants
Mathematical analysis
Mathematical models
Multiple orthogonal polynomials
Poles
Riemann surfaces
Uniformization
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Summary:An algebraic function of the third order plays an important role in the problem of asymptotics of Hermite–Padé approximants for two analytic functions with branch points. This algebraic function appears as the Cauchy transform of the limiting measure of the asymptotic distribution of the poles of the approximants. In many cases this algebraic function can be determined by using the given position of the branch points of the functions which are approximated and by the condition that its Abelian integral has purely imaginary periods. In the present paper we obtain a hyperelliptic uniformization of this algebraic function. In the case when each approximated function has only two branch points, the genus of this function can be equal to 0, 1 (elliptic case) or 2 (ultra-elliptic case). We use this uniformization to parametrize the elliptic case. This parametrization allows us to obtain a numerical procedure for finding this elliptic curve and as a result we can describe the limiting measure of the distribution of the poles of the approximants.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2014.11.048