Algebroid functions, Wirsing's theorem and their relations
type: Item
instance of: scholarly article
title: Algebroid functions, Wirsing's theorem and their relations
zbMATH Open document ID: 0951.32015
DOI: 10.1007/PL00004787
author: Min Ru
published in: Mathematische Zeitschrift
publication date: 2000-12-18T00:00:00Z
review text: The author deals with the relation between Nevanlinna theory and Diophantine approximation. The author first points out a relationship between the Second Main Theorem for algebroid functions in Nevanlinna theory and Wirsing's theorem [\textit{E. Wirsing}, Proc. Sympos. Pure Math. 20, 213-247 (1971; Zbl 0223.10017)] in Diophantine approximation. The second part of this paper deals with moving target problem for holomorphic mappings into compact Riemann surfaces. Its counterpart in Diophantine approximation follows from a recent work of J. Tucker (preprint pp. 30). Indeed, Tucker actually proved a more general result. The author also deals with the higher dimensional case. In this part the author follows Stoll's method of algebroid reduction to obtain a weak version of the Second Main Theorem on a finite analytic covering space over \(\pi:M\to \mathbb{C}^m\). Related to this, the author mentions that the counting function \(N(r,\beta)\) of the ramification divisor of \(\pi:M\to \mathbb{C}^m\) is estimated by the characteristic function of a meromorphic mapping \(M\to\mathbb{P}(V)\) if \(f\) separates the fibers of \(\pi:M\to \mathbb{C}^m\). However, this assumption on the meromorphic mapping \(f\), that is, \(f\) separates the fibers of \(\pi:M\to \mathbb{C}^m\), does not need to obtain the estimate on \(N(r,\beta)\). Indeed, this estimate always holds for nonconstant meromorphic mappings \(f:M\to \mathbb{P}(V)\) by a theorem due to \textit{J. Noguchi} [Hiroshima Math. J. 6, 265-280 (1976; Zbl 0338.32016)]. The author finally gives a conjecture of Wirsing's version of Schmidt's subspace theorem.
reviewed by: Yoshihiro Aihara
Mathematics Subject Classification ID: 32H30
Mathematics Subject Classification ID: 11J97
Mathematics Subject Classification ID: 32A22
Mathematics Subject Classification ID: 11J68
zbMATH DE Number: 1406452
zbMATH Keywords: Schmidt's subspace theorem
zbMATH Keywords: Nevanlinna theory
zbMATH Keywords: algebroid functions
zbMATH Keywords: Wirsing's theorem
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: moving targets
MaRDI profile type: Publication
Recommended article: Schmidt's subspace theorem for moving hypersurface targets
Recommended article: Q4332780
Recommended article: A Cartan's second main theorem approach in Nevanlinna theory
Recommended article: The second main theorem with moving targets on parabolic manifolds
Recommended article: Diophantine Approximation and Nevanlinna Theory
Recommended article: Second main theorem and uniqueness theorem with moving targets on parabolic manifolds
Recommended article: Q4793642
Recommended article: Schmidt's subspace theorem with moving targets
Recommended article: The second main theorem for moving targets
Recommended article: Quantitative subspace theorem and general form of second main theorem for higher degree polynomials
description: scientific article; zbMATH DE number 1406452