Équations exponentielles-polynômes et suites récurrentes linéaires. II. (Exponential polynomial equations and linear recurrence sequences. II)
type: Item
instance of: scholarly article
title: Équations exponentielles-polynômes et suites récurrentes linéaires. II. (Exponential polynomial equations and linear recurrence sequences. II)
zbMATH Open document ID: 0661.10027
DOI: 10.1016/0022-314X(89)90050-4
author: Michel Laurent
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: Let C be a field of characteristic 0, r a positive integer, L a finite set of subscripts, \(P_{\ell}(X)\in C[X_ 1,...,X_ r]\) for \(\ell \in L\), and \(\chi_{\ell}(X)\) (\(\ell \in L)\) expressions of the form \(b^{X_ 1}_{\ell_ 1}...b^{X_ r}_{\ell_ r}\), with \(b_{\ell 1},...,b_{\ell r}\in C^*\). Partition L into pairwise disjoint subsets \(L_ 1,...,L_ m\). The author considers systems of so- called exponential-polynomial equations \[ (1)\quad \sum_{\ell \in L_ i}P_{\ell}(x)\chi_{\ell}(x)=0\quad in\quad x\in {\mathbb{Z}}^ r \] for \(i=1,...,m\). By a partition \({\mathcal P}\) of \((L_ 1,...,L_ m)\) we mean a collection of pairwise disjoint non-empty sets \(L_{ij}\) \((i=1,...,m\), \(j\in J_ i)\) such that \(L_ i=\cup_{j\in J_ i}L_{ij}\) for \(i=1,...,m\). A solution x of (1) is called \({\mathcal P}\)-compatible if \[ (2)\quad \sum_{\ell \in L_{ij}}P_{\ell}(x)\chi_{\ell}(x)=0\quad for\quad i=1,...,m,\quad j\in J_ i \] and if (2) does not remain valid if \({\mathcal P}\) is replaced by a finer partition. Let \(H_{{\mathcal P}}\) be the subgroup of \({\mathbb{Z}}^ r\) defined by \(\chi_{\ell}(x)=\chi_{\ell '}(x)\) for each pair \(\ell,\ell '\) belonging to the same subset \(L_{ij}\) for \(i=1,...,m\), \(j\in J_ i\). Let \(S_{{\mathcal P}}\) be the set of \({\mathcal P}\)-compatible solutions of (1) such that \(P_{\ell}(x)\neq 0\) for each \(\ell \in L\). For \(x=(x_ 1,...,x_ r)\in {\mathbb{Z}}^ r\) put \(| x| =\max (| x_ 1|,...,| x_ r|)\). The author's main result is as follows: Theorem 1. Every \(x\in S_{{\mathcal P}}\) can be expressed as \(x'+x''\), where \(x''\in H_{{\mathcal P}}\) and \(| x'| \leq C\) if all polynomials \(P_{\ell}\) are constant and \(| x'| \leq C \log | x|\) otherwise, where C is a constant independent of x. In an earlier paper, the author proved this result for the case that C is an algebraic number field [Invent. Math. 78, 299-327 (1984; Zbl 0554.10009)]. The author obtains the general result by applying a result of \textit{W. D. Brownawell} …
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11B37
zbMATH DE Number: 4079523
zbMATH Keywords: decomposition theorem
zbMATH Keywords: common zeros
zbMATH Keywords: exponential-polynomial equations
zbMATH Keywords: linear recurrence sequences
reviewed by: Jan-Hendrik Evertse
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4079523