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

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 $F_{\hat{I}}»(X,Y)=\hat{I}¾$, where $\hat{I}¾\in {\mathbb{C}}^{\times }$ and \begin{equation*} F_λ(X,Y):=X^4 -λX^3Y -6 X^2Y^2 + λXY^3 +Y^4, \quad \quad λ\in \mathbb{C}[T]/\{\mathbb{C}\} \end{equation*} denotes a family of quartic simple forms.

19 pages. Minor revisions including theorem relabeling. Accepted for publication in the Rocky Mountain Journal of Mathematics