Thue equations over $\mathbb{C}(T)$: The Complete Solution of a Simple quartic family

TL;DR

This paper completely solves a family of quartic Thue equations over the function field T, using the ABC-Theorem to find all polynomial solutions for a specific simple quartic family.

Contribution

It provides the first complete solution to a simple quartic family of Thue equations over T, applying the ABC-Theorem to characterize all solutions.

Findings

All solutions TT to the family of Thue equations are explicitly determined.

The ABC-Theorem is effectively applied to polynomial equations over T.

The solutions depend on the parameter T, with explicit formulas derived.

Abstract

In this paper we completely solve a simple quartic family of Thue equations over $\mathbb{C}(T)$. Specifically, we apply the ABC-Theorem to find all solutions $(x,y) \in \mathbb{C}[T] \times \mathbb{C}[T]$ to the set of Thue equations $F_{\lambda}(X,Y) = \xi$, where $\xi \in \mathbb{C}^{\times}$ and \begin{equation*} F_{\lambda}(X,Y):=X^4 -\lambda X^3Y -6 X^2Y^2 + \lambda XY^3 +Y^4, \quad \quad \lambda \in \mathbb{C}[T]/\{\mathbb{C}\} \end{equation*} denotes a family of quartic simple forms.