The equation \(ax+by=1\) in characteristic \(p\)
type: Item
instance of: scholarly article
title: The equation \(ax+by=1\) in characteristic \(p\)
zbMATH Open document ID: 0916.11019
author: José Felipe Voloch
published in: Journal of Number Theory
publication date: 1999-07-08T00:00:00Z
review text: Let \(K\) be a field and \(G\) a finite rank subgroup of the multiplicative group \(K^* \times K^*\). Consider the equation (*) \(ax+by=1\) with \(a,b\in K^*\) to be solved in \((x,y)\in G\). In case that \(K\) has characteristic \(0\), Schlickewei showed that the number of solutions of (*) is bounded by a quantity depending only on the rank \(r\) of \(G\) and later \textit{F. Beukers} and \textit{H. Schlickewei} obtained the upper bound \(2^{8r+16}\) [Acta Arith. 78, 189-199 (1996; Zbl 0880.11034)]. The proofs of Schlickewei and Beukers and Schlickewei use diophantine approximation techniques which can not be carried over to positive characteristic. Furthermore, equation (*) need not even have finitely many solutions if the characteristic \(p\) of \(K\) is positive. For instance if \(a=b=1\) and \((x,y)\) is a solution of (*) then so is \((x^{p^n},y^{p^n})\) for every positive integer \(n\). The author proves the following result. Suppose that \(K\) has characteristic \(p>0\). Let \(G\) be a finite type subgroup of \(K^*\times K^*\) of rank \(r\). Then (*) has at most \(p^r(p^r+p-2)/(p-1)\) solutions \((x,y)\in G\) unless \((a^n,b^n)\in G\) for some \(n\geq 1\). The author gives an elementary proof. At the end of his paper he discusses some special cases of equation (*) in which the number of solutions can be bounded independently of the characteristic \(p\).
reviewed by: Jan-Hendrik Evertse
Mathematics Subject Classification ID: 11D57
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 1247935
zbMATH Keywords: diophantine equations over characteristic \(p\)
zbMATH Keywords: number of solutions
MaRDI profile type: Publication
full work available at URL: jnth.1998.2295
OpenAlex ID: W1598276965
cites work: Q4016340
cites work: Uniform bound for generic points of curves in tori.
cites work: The equation x+y=1 in finitely generated groups
cites work: Q4893917
cites work: The Mordell-Lang conjecture for function fields
DOI: 10.1006/JNTH.1998.2295
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description: scientific article; zbMATH DE number 1247935