Diophantine equations over global function fields. I: The Thue equation
type: Item
instance of: scholarly article
title: Diophantine equations over global function fields. I: The Thue equation
zbMATH Open document ID: 1157.11011
published in: Journal of Number Theory
publication date: 2006-11-15T00:00:00Z
review text: In this paper, the authors devise an algorithm to solve Thue equations over global function fields. Let \(k=\mathbb{F}\) be a finite field with \(q=p^d\) elements, \(k(t)\) the rational function field of \(k\), and \(K\) a finite extension of \(k(t)\) of degree \(n_0\) and genius \(g_0\). The integral closure of \(K[t]\) is denoted by \(o_K\). A Thue equation over \(k\) is an equation of the form \[ F(x, y)=m\;\; \text{in } x, y \in o_K \] where \[ F(X, Y) =\sum_{i=0}^{n} A_iX^{n-i}Y^n\,\in o_K[X, Y] \] is a binary homogeneous form of degree at least \(3\). Let \(\alpha\) be a zero of \(F(x, 1)\) in \(\overline{K}\) and \(L=K(\alpha)\). The Thue equation becomes \[ F(x, y)=N_{L/K}(x-\alpha y)=m,\;\; x, y \in o_K. \] The authors show all steps for solving this norm equation using a method similar to that in algebraic number field (S-units, Dirichlet's unit theorem, Siegel's identity,\dots). They also adapt Mason's inequality on height to the situation. To illustrate the efficiency of this algorithm, the authors finish the paper with four examples of Thue equations over the function field \(K=k(t)\), where \(k=\mathbb{F}_q\) with \(q=3, 5, 11\). Later, the authors improved the algorithm so it can completely determine all solutions and be more more efficient, see the review of Part II [Exp. Math. 15, No. 1, 1--6 (2006; Zbl 1142.11019)].
reviewed by: Alain Togbé
Mathematics Subject Classification ID: 11D59
Mathematics Subject Classification ID: 11Y50
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 5072782
zbMATH Keywords: Thue equations
zbMATH Keywords: Global function fields
author: István Gaál
author: Michael E. Pohst
describes a project that uses: KANT/KASH
MaRDI profile type: Publication
full work available at URL: j.jnt.2005.10.009
OpenAlex ID: W1995956684
cites work: Q3998726
cites work: KANT V4
cites work: Q4359498
cites work: Q4535933
cites work: Bounds for the solutions of norm form, discriminant form and index form equations in finitely generated integral domains
cites work: Q3315578
cites work: Rational points on curves over finite fields. Theory and Applications
cites work: Q4170806
cites work: Algebraic function fields and codes
cites work: Q5723265
DOI: 10.1016/J.JNT.2005.10.009
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description: scientific article; zbMATH DE number 5072782