Diophantine equations over global function fields. I: The Thue equation
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)].
scientific article; zbMATH DE number 5072782
scientific article