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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 2024 Volume 215, Number 12, Pages 56–88 (Mi sm10114)

This article is cited in 1 paper

Zeros of discriminants constructed from Hermite–Padé polynomials of an algebraic function and their relation to branch points

A. V. Komlov, R. V. Palvelev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: Let $f_\infty$ be the germ at $\infty$ of some algebraic function $f$ of degree $m+1$. Let $Q_{n,j}$, $j=0,\dots,m$, be the Hermite–Padé polynomials of the first type of order $n\in\mathbb N$ constructed from the tuple of germs $[1, f_ \infty, f_\infty^2,\dots,f_\infty^m]$. We study the asymptotic properties of discriminants constructed from the Hermite–Padé polynomials in question, that is, the discriminants $D_n(z)$ of the polynomials $Q_{n,m}(z)w^m+Q_{n,m-1}(z)w^{m-1}+\dots+Q_{n,0}(z)$. We find their weak asymptotics, as well as the asymptotic behaviour of their ratio with the polynomial $Q_{n,m}^{2m-2}$. In addition, we refine the weak asymptotic formulae for $D_n$ at branch points of the original algebraic function $f$ and apply the results obtained to the problem of finding branch points of $f$ numerically on the basis of the prescribed germ $f_\infty$, which is used in applied problems.
Bibliography: 49 titles.

Keywords: Hermite–Padé polynomials, discriminants, branch points, algebraic functions, weak asymptotics.

MSC: Primary 41A21; Secondary 14H30, 30C15

Received: 09.05.2024 and 29.08.2024

DOI: 10.4213/sm10114


 English version:
Sbornik: Mathematics, 2024, 215:12, 1633–1665

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© Steklov Math. Inst. of RAS, 2025