On the Eisenstein constant
type: Item
instance of: scholarly article
title: On the Eisenstein constant
zbMATH Open document ID: 1165.11083
author: Rachid Mechik
published in: Annales Mathématiques Blaise Pascal
publication date: 2008-08-14T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10555
review text: Eisenstein proved that if an entire series \(u=\sum_{n\geq0} \alpha_nz^n\) with rational coefficients represents an algebraic function, then there exists an integer \(a\) (called an Eisenstein constant) such that \(a^n\alpha_n\) is an integer for all \(n\). Precise values for Eisenstein constants, or upper bounds for them, are known in a relatively small number of cases [see for instance \textit{W. M. Schmidt}, ``Eisenstein's theorem on power series expansions of algebraic functions'', Acta Arith. 56, 161--179 (1990; Zbl 0659.12003) and \textit{B. M. Dwork} and \textit{A. J. van der Poorten}, ``The Eisenstein constant'', Duke Math. J. 65, 23--43 (1992; Zbl 0770.11051)]. The main result of the paper is a criterion which allows to produce Eisentein constants for regular algebraic functions. The method is explained in detail for functions of type \((1-x)^c\), where \(c\) is a positive rational mumber.
reviewed by: Roberto Dvornicich
Mathematics Subject Classification ID: 11S05
Mathematics Subject Classification ID: 13J05
Mathematics Subject Classification ID: 12J25
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 5312017
zbMATH Keywords: algebraic functions
zbMATH Keywords: Eisenstein constant
MaRDI profile type: Publication
OpenAlex ID: W2526799785
cites work: Algebraic power series and diagonals
cites work: Q3244542
cites work: The Eisenstein constant
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cites work: On Natural Radii of p-Adic Convergence
cites work: Algebraic functions over finite fields
cites work: Eisenstein's theorem on power series expansions of algebraic functions
DOI: 10.5802/AMBP.241
description: scientific article; zbMATH DE number 5312017