Generalized Büchi's problem for algebraic functions and meromorphic functions
type: Item
instance of: scholarly article
title: Generalized Büchi's problem for algebraic functions and meromorphic functions
zbMATH Open document ID: 1347.11033
author: Ta Thi Hoai An
author: Hsiu-Lien Huang
author: Julie Tzu-Yueh Wang
published in: Mathematische Zeitschrift
publication date: 2013-02-06T00:00:00Z
review text: The paper under review analyzes Büchi\'{}s problem for \(n\)-th powers over function fields of characteristic 0 and fields of meromorphic functions. Büchi's problem was originally stated for \(n=2\) and for rings of integers. More specifically the question concerned sequences of integers \(\{x_k\}\) satisfying \(x_{k+2}^2-2x_{k+1}^2 + x_k^2=2\) and existence of a constant \(M\) such that any sequence as described above of length greater or equal to \(M\) must be of the form \(x_n=x +n\) for some \(x \in \mathbb Z\). This problem is still open. The positive answer to this question would lead to a stronger version of the negative answer to Hilbert's Tenth Problem (the question of decidability of Diophantine equations). The original problem has been generalized in several ways: the squares have been replaced by an arbitrary power \(n\) with the condition that the \(n\)-th difference is equal to \(n!\), and the ring of integers has been replaced by other commutative rings. Function field generalizations of this problem have been obtained by \textit{H. Pasten} [Proc. Am. Math. Soc. 138, No. 5, 1549--1557 (2010; Zbl 1222.11146); J. Number Theory 133, No. 9, 2964--2998 (2013; Zbl 1364.11090)], [\textit{T. Pheidas} and \textit{X. Vidaux}, J. Lond. Math. Soc., II. Ser. 74, No. 3, 545--565 (2006; Zbl 1109.03032); corrigendum 82, No. 1, 273--278 (2010); Pac. J. Math. 238, No. 2, 349--366 (2008; Zbl 1151.03004)], [\textit{A. Shlapentokh} and \textit{X. Vidaux}, J. Algebra 330, No. 1, 482--506 (2011; Zbl 1250.11103)], [\textit{P. Vojta}, Contemp. Math. 270, 261--274 (2000; Zbl 0995.11070)] and [\textit{J. T. Y. Wang}, Int. J. Number Theory 8, No. 2, 507--524 (2012; Zbl 1261.11035)], [\textit{T. T. H. An} and \textit{J. T. Y. Wang}, J. Math. Anal. Appl. 381, No. 2, 661--677 (2011; Zbl 1246.30076)], and others. In this paper, the generalized problem is considered and solved over rings and fields of meromorphic and algebraic functions of characteristic 0. The lower bound on the con…
reviewed by: Alexandra Shlapentokh
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11U05
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 11J97
zbMATH DE Number: 6134696
zbMATH Keywords: Büchi's problem
MaRDI profile type: Publication
full work available at URL: s00209-012-0997-9
OpenAlex ID: W2004875184
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DOI: 10.1007/S00209-012-0997-9
description: scientific article; zbMATH DE number 6134696