Abstract

Let \(\Omega\) be a simple-connected domain, \(Z=\{z_{k}\in\Omega,\ k=\overline{1,N}\}\ (N\leq\infty)\), and let \(TM(\Omega,Z)\) be a set of functions meromorphic in \(\Omega\) and satisfying at each its pole \(z_{k}\) the trivial monodromy condition. The criterion for trivial monodromy is well known (it was established by Duistermaat and Grünbaum in 1987). However, this criterion is of a local nature. It is impossible to extract from it any information about the structure of the set \(TM(\Omega,Z)\). In this paper, we obtaine an explicit description of the set \(TM(\Omega,Z)\) for \(N<\infty\). In the case \(N=\infty\), we establishe a certain analogue of the Mittag-Leffler theorem: for any sequence of natural numbers \(\{m_{k}\}\) there exists a function \(q\in TM(\Omega,Z)\), which at each point \(z_{k}\) satisfies the Duistermaat–Grünbaum condition with indicator \(m_{k}\).

中文翻译：

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具有平凡单一性的势类

摘要

令\(\Omega\)为单连通域，\(Z=\{z_{k}\in\Omega,\ k=\overline{1,N}\}\ (N\leq\infty)\ )，让\(TM(\Omega,Z)\)是一组在\(\Omega\)中亚纯的函数，并且在它的每个极点\(z_{k}\)满足平凡的单调条件。琐碎单一性的标准是众所周知的（它由 Duistermaat 和 Grünbaum 于 1987 年建立）。然而，这个标准是地方性的。不可能从中提取任何关于集合\(TM(\Omega,Z)\) 结构的信息。在本文中，我们获得了\(N<\infty\)集合\(TM(\Omega,Z)\)的显式描述。在这种情况下\(N=\infty\)，我们建立了 Mittag-Leffler 定理的某个类比：对于任何自然数序列\(\{m_{k}\}\)存在一个函数\(q\in TM( \Omega,Z)\)，在每个点\(z_{k}\)满足 Duistermaat-Grünbaum 条件，指标为\(m_{k}\)。