Elementary resolution of a family of quartic Thue equations over function fields

TL;DR

This paper completely solves a family of quartic Thue equations over complex function fields, providing elementary and short proofs by analyzing a smaller ring where units are easier to determine.

Contribution

It offers a novel elementary approach to solving parametrized quartic Thue equations over function fields, avoiding height bounds and simplifying previous complex methods.

Findings

Explicit solutions for the family of equations are determined.

The method simplifies the proof process compared to previous approaches.

The approach can be applied to similar problems over function fields.

Abstract

We consider and completely solve the parametrized family of Thue equations \begin{eqnarray*}X(X-Y)(X+Y)(X-\lambda Y)+Y^4=\xi,\end{eqnarray*} where the solutions $x,y$ come from the ring $\mathbb{C}[T]$, the parameter $\lambda\in\mathbb{C}[T]$ is some non-constant polynomial and $0\neq\xi\in\mathbb{C}$. It is a function field analogue of the family solved by Mignotte, Peth\H{o} and Roth in the integer case. A feature of our proof is that we avoid the use of height bounds by considering a smaller relevant ring for which we can determine the units more easily. Because of this, the proof is short and the arguments are very elementary (in particular compared to previous results on parametrized Thue equations over function fields).