The generalized Fermat equation in function fields
type: Item
instance of: scholarly article
title: The generalized Fermat equation in function fields
zbMATH Open document ID: 0739.11012
DOI: 10.1016/0022-314X(91)90053-E
author: Julia Mueller
author: Enrico Bombieri
published in: Journal of Number Theory
publication date: 1992-06-26T00:00:00Z
review text: Let \(k\) be an algebraically closed field of characteristic 0 and \(K=k(t)\) a function field over \(k\). Let \(a_ i\) (\(i=1,\dots,n\)) be fixed elements in \(K\) (not all zero) and consider the solutions \(z_ 1,\dots,z_ n\in K\) of the generalized Fermat equation \[ a_ 1z^ r_ 1+\dots+a_ nz^ r_ n=0. \tag{1} \] In case of the basic Fermat equation (\(n=3\)) \textit{J. H. Silverman} [Trans. Am. Math. Soc. 273, 201-205 (1982; Zbl 0493.10020)] and \textit{J. F. Voloch} [Bol. Soc. Bras. Mat. 16, No. 2, 29-39 (1985; Zbl 0612.10011)] gave upper bounds for the height and the number of solutions of the equation as a function of \(r\) and the heights of the coefficients \(a_ i\). On the other hand \textit{J. Mueller} [Compos. Math. 73, 189-197 (1990; Zbl 0697.10015)] proved, that if \(r>30\), then a ``non- degenerate'' equation \(ax^ r+by^ r=c\) can have at most two non-zero solutions \(x,y\in K\). This bound is best possible and independent of the coefficients \(a,b,c\in K\). In the present paper the authors extend the above mentioned result of J. Mueller to the generalized Fermat equation (1), namely they prove, that if \(r>n!(n!-2)\), then the solution of (1) forms at most \(n!^{n!}\) families of solutions, where the families have an explicitly given simple structure. The main tool of the proof is the function field analogue of the \(abc\)-inequality.
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 23713
zbMATH Keywords: abc-conjecture
zbMATH Keywords: function field
zbMATH Keywords: generalized Fermat equation
zbMATH Keywords: families of solutions
reviewed by: István Gaál
MaRDI profile type: Publication
cites work: Vanishing sums in function fields
cites work: Remark on two theorems of Siegel
cites work: Q3474708
cites work: Q3315578
cites work: Q4135685
cites work: The Catalan Equation over Function Fields
cites work: Diagonal equations over function fields
full work available at URL: 0022-314x(91)90053-e
OpenAlex ID: W2008147755
description: scientific article; zbMATH DE number 23713