Thomas’ conjecture over function fields

Ziegler, Volker

Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 289-309.

Thomas’ conjecture is, given monic polynomials $p_1,$ $...,p_d\in \mathbb{Z}\left[a\right]$ with $0<deg{p}_1<\cdots <deg{p}_d$, then the Thue equation (over the rational integers)

$$(X-p_1\left(a\right)Y)\cdots (X-p_d\left(a\right)Y)+Y^d=1$$

has only trivial solutions, provided $a\ge a_0$ with effective computable $a_0$. We consider a function field analogue of Thomas’ conjecture in case of degree $d=3$. Moreover we find a counterexample to Thomas’ conjecture for $d=3$.

La conjecture de Thomas affirme que, pour des polynômes unitaires $p_1,...,p_d\in \mathbb{Z}\left[a\right]$ tels que $0<deg{p}_1<\cdots <deg{p}_d$, l’équation de Thue

$$(X-p_1\left(a\right)Y)\cdots (X-p_d\left(a\right)Y)+Y^d=1$$

n’admet pas de solution non triviale (dans les entiers relatifs) pourvu que $a\ge a_0$, avec une borne effective $a_0$. Nous nous intéressons à un analogue de la conjecture de Thomas sur les corps de fonctions pour le degré $d=3$ et en donnons un contrexemple.

Keywords: Thue equation, function fields

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