Diophantine Equations over Global Function Fields II: R-Integral Solutions of Thue Equations

Abstract

Let K be an algebraic function field over a finite field. Let L be an extension field of K of degree at least 3. Let R be a finite set of valuations of K and denote by S the set of extensions of valuations of R to L. Denote by O_K,R, O_L,S the ring of Rintegers of K and S-integers of L, respectively. Assume that α ∈ O_L,S with L = K(α), let 0 ≠ µ ∈ O_K,R, and consider the solutions (x, y) ∈ O_K,R of the Thue equation

N_L/K (x - αy) = µ.

We give an efficient method for calculating the R-integral solutions of the above equation. The method is different from that in our previous paper [Gaál and Pohst 06] and is much more efficient in many cases.

Keywords:

• Primary 11D59, 11Y50, 11R58
• Thue equations
• global function fields