Linear equations over multiplicative groups in positive characteristic
type: Item
instance of: scholarly article
title: Linear equations over multiplicative groups in positive characteristic
zbMATH Open document ID: 1292.11053
author: Dominik J. Leitner
published in: Acta Arithmetica
publication date: 2012-05-11T00:00:00Z
Mathematics Subject Classification ID: 11D04
Mathematics Subject Classification ID: 11D72
zbMATH DE Number: 6033771
zbMATH Keywords: linear equations
zbMATH Keywords: equations in many variables
zbMATH Keywords: S-unit equations
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4064/aa153-4-1
OpenAlex ID: W2001857794
DOI: 10.4064/AA153-4-1
review text: \(S\)-unit equations and related equations over number fields or over function fields of characteristic \(0\), as well as generalizations in the vain of the Mordell-Lang conjecture for semi-abelian varieties, can be treated by techniques from Diophantine approximation, but these techniques are not applicable for fields of positive characteristic. In the past few years, various new techniques have been developed to treat Diophantine equations in positive characteristic, based on abc-type inequalities for function fields of positive characteristic, differentiation of algebraic functions, model theory and finite automata. Building further on earlier individual work of both authors, \textit{H. Derksen} and \textit{D. Masser} [Proc. Lond. Math. Soc. (3) 104, No. 5, 1045--1083 (2012; Zbl 1269.11062)] proved a general result concerning the structure of sets \(V\cap G^n\), where \(V\) is a linear subvariety of \(K^n\) for some field \(K\) of characteristic \(p>0\), \(G\) is a finitely generated subgroup of \(K^*\) and \(G^n\subset (K^*)^n\) is the \(n\)-fold direct product of \(G\). Their argument is elementary, and makes essential use of differentiation of algebraic functions. In the paper under review, the author gives more detailed results in some special cases.NEWLINENEWLINETo describe the above mentioned results, we introduce some further notation. For any power \(q\) of \(p\), we denote by \(\varphi_q\) the Frobenius map \((x_1,\dots , x_n)\mapsto (x_1^q,\dots ,x_n^q)\) on \(K^n\). A \(G\)-automorphism of \(K^n\) is a map \((x_1,\dots ,x_n)\mapsto (g_1x_1,\dots ,g_nx_n)\), where \(g_1,\dots ,g_n\in G\). For any tuple of \(G\)-automorphisms \(\psi_1,\dots ,\psi_h\) we define the operator NEWLINE\[NEWLINE [\psi_1,\dots ,\psi_h]_q :=\bigcup_{e_1=0}^{\infty}\cdots\bigcup_{e_h=0}^{\infty} (\psi_1^{-1}\varphi_q^{e_1}\psi_1)\cdots (\psi_h^{-1}\varphi_q^{e_h}\psi_h) NEWLINE\]NEWLINE which maps each point of \(K^n\) to an infinite subset of \(K^n\); here the operator is just t…
reviewed by: Jan-Hendrik Evertse
description: scientific article; zbMATH DE number 6033771