Thue equations over $\mathbb{C}(T)$: The Complete Solution of a Simple quartic family
Bernadette Faye, Ingrid Vukusic, Ezra Waxman, Volker Ziegler
TL;DR
This work solves a simple quartic family of Thue equations over the function field $ ext{C}(T)$ by applying the function-field ABC theorem to bound heights of associated $S$-units. The authors analyze the unit group of the extension $ ext{C}[T][oldsymbol extα]$, where $oldsymbol extα$ is a root of $f_oldsymbol extλ$, and show that the units are generated by $oldsymbol extα-1$, $oldsymbol extα$, and $oldsymbol extα+1$. Using these bounds, they reduce the problem to a finite search, ultimately proving that the solution set to $F_oldsymbol extλ(X,Y)=oldsymbol extξ$ consists precisely of $(x,y)=(oldsymbol extη,0),(0,oldsymbol extη)$ with $oldsymbol extη^4=oldsymbol extξ$ and $(x,y)=(oldsymbol extη,oldsymbol extη),(oldsymbol extη,-oldsymbol extη)$ with $-4oldsymbol extη^4=oldsymbol extξ$, up to units. The method highlights the effectiveness of function-field ABC bounds compared to Baker-type bounds and is complemented by an implementable Sage routine. The results provide a complete, explicit resolution for this quartic family and demonstrate a scalable approach for similar Thue equations over function fields.
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 $F_λ(X,Y) = ξ$, where $ξ\in \mathbb{C}^{\times}$ 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.