Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better
type: Item
instance of: scholarly article
title: Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better
zbMATH Open document ID: 0575.10032
DOI: 10.1016/0022-314X(85)90061-7
author: Charles F. Osgood
published in: Journal of Number Theory
publication date: 1985-01-01T00:00:00Z
review text: [A short version of this paper has been published in Contemp. Math. 25, 129-130 (1983; Zbl 0528.30020).] This paper proves two results by a common construction. The first result is in Nevanlinna Theory. A fundamental result due to Picard says that a nonconstant meromorphic function assumes every value with at most two exceptions. In the 20th century Nevanlinna showed that there is an asymptotic bound on the number of solutions of \(f(z)=a\) in circles centered about \(z=0\), with equality holding for all but an at most countable set of a's. For those values of a for which this asymptotic bound is not attained he defined a number called the defect that can range between 0 and 1. Nevanlinna showed that the sum of all of the defects of a nonconstant meromorphic function is at most 2. Nevanlinna conjectured but could not show that if the set of possible a values was enlarged to include very small growth meromorphic functions the bound of 2 would still hold. In this paper the author gives the first proof of Nevanlinna's conjecture with no unnatural restrictions on the large meromorphic function. The paper is lengthy in part because of the method of proof and in part because of the number of results that it establishes in both complex variables and number theory. His Theorem III is a generalization of the n small function theorem to linear forms. That it really is a generalization is established with equation (21). The number-theoretic parts of this paper deal with obtaining effectively computable lower bounds on the approximation of certain types of formal power series. A typical result bounds the goodness of the approximation of formal power series solutions to linear differential equations having polynomial coefficients, by rational functions. The measure of goodness of such approximations is taken to be the number of places of exact agreement between the two functions at a set of chosen points in the extended plane. Here the rational function plays the part of the lar…
Mathematics Subject Classification ID: 11J99
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 34A45
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 12H20
zbMATH DE Number: 3918476
zbMATH Keywords: Nevanlinna Theory
zbMATH Keywords: meromorphic functions
zbMATH Keywords: first proof of Nevanlinna's conjecture
zbMATH Keywords: generalization of the n small function theorem
zbMATH Keywords: effectively computable lower bounds
zbMATH Keywords: approximation of formal power series
zbMATH Keywords: solutions to linear differential equations
zbMATH Keywords: rational functions
zbMATH Keywords: algebraic differential equation
zbMATH Keywords: Thue-\(Siegel\)-\(Roth\)-\(Schmidt\) Theorem
reviewed by: Fred Gross
MaRDI profile type: Publication
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full work available at URL: 0022-314x(85)90061-7
OpenAlex ID: W2089636200
description: scientific article; zbMATH DE number 3918476