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arXiv · 2301.06129

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

Bernadette Faye·Ingrid Vukusic·Ezra Waxman·Volker Ziegler
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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.

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Bernadette Faye, Ingrid Vukusic, Ezra Waxman, Volker Ziegler. 2023-01-15. Thue equations over $\mathbb{C}(T)$: The Complete Solution of a Simple quartic family. https://doi.org/10.1216/rmj.2025.55.1263

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