On the numbers of solutions of decomposable polynomial equations
type: Item
instance of: scholarly article
title: On the numbers of solutions of decomposable polynomial equations
zbMATH Open document ID: 0655.10017
DOI: 10.1007/BF01194408
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: In a recent paper \textit{J. H. Evertse} and \textit{K. Györy} [Acta Arith. 50, No.4, 357-379 (1988; Zbl 0595.10013)] gave criteria for the finiteness of the numbers of solutions of decomposable form equations (including Thue equations, norm form equations, discriminant form equations and index form equations) over finitely generated integral domains over \({\mathbb{Z}}\). On the other hand \textit{I. Gaál} considered inhomogeneous extensions of certain classes of decomposable form equations [cf. Publ. Math. 34, 109-122 (1987; Zbl 0626.10015)]. In the present paper the authors give criteria for the finiteness of the numbers of solutions of decomposable polynomial equations (i.e. ``inhomogeneous'' decomposable form equations) over fis S(d,c): For each continuous function f on [-1,1]\(\times ({\mathbb{R}}/{\mathbb{Z}})\) \[ \lim_{N\to \infty}(1/N)\sum^{N}_{c=1}(1/c)\sum_{d mod c, (d,c)=1}f(S(d,c)/c, d/c)=(6/\pi^ 2)\int^{1}_{0}f(0,t) dt \] and \[ \lim_{N\to \infty}(1/N)\sum^{N}_{c=1}\sum_{d mod c, (d,c)=1}(S(d,c)/c)^ 2 f(S(d,c)/c, d/c)= \] \[ \sum^{\infty}_{n=1}\sum_{d| n}\beta (n/d)(1/d)\sum_{m mod d}(1/12n)^ 2\{f(1/12n, m/d)+f(-1/12n, m/d)\} \] with \(\beta\) (k) defined by \(\sum^{\infty}_{k=1}\beta (k)k^{- s}=\zeta (s) \zeta (s+1)^{-2}\) for Re s\(>1\).
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11D57
zbMATH DE Number: 4068977
zbMATH Keywords: integral domains
zbMATH Keywords: inhomogeneous Thue equations
zbMATH Keywords: finiteness criteria
zbMATH Keywords: bounds for the number of solutions
zbMATH Keywords: decomposable polynomial equations
zbMATH Keywords: decomposable form equations
author: Jan-Hendrik Evertse
author: István Gaál
author: Kálmán Győry
reviewed by: Roelof Bruggeman
MaRDI profile type: Publication
cites work: Q3029071
cites work: Bounds for the solutions of norm form, discriminant form and index form equations in finitely generated integral domains
cites work: Q3309989
cites work: Exponential diophantine equations
cites work: A General Theory of Algebraic Geometry Over Dedekind Domains, I: The Notion of Models
cites work: The \(\mathfrak p\)-adic Thue-Siegel-Roth-Schmidt theorem
cites work: On norm form equations
cites work: Diophantine approximation
full work available at URL: https://doi.org/10.1007/bf01194408
OpenAlex ID: W2615924184
description: scientific article; zbMATH DE number 4068977