Thomas’ conjecture over function fields
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 [a] with 0<degp 1 <<degp d , then the Thue equation (over the rational integers)

(X-p1(a)Y)(X-pd(a)Y)+Yd=1

has only trivial solutions, provided aa 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 [a] tels que 0<degp 1 <<degp d , l’équation de Thue

(X-p1(a)Y)(X-pd(a)Y)+Yd=1

n’admet pas de solution non triviale (dans les entiers relatifs) pourvu que aa 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.

Received: 2005-11-16
Published online: 2008-12-03
DOI: https://doi.org/10.5802/jtnb.587
Keywords: Thue equation, function fields
@article{JTNB_2007__19_1_289_0,
     author = {Volker Ziegler},
     title = {Thomas' conjecture over function fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     pages = {289-309},
     doi = {10.5802/jtnb.587},
     zbl = {pre05186988},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2007__19_1_289_0/}
}
Ziegler, Volker. Thomas’ conjecture over function fields. Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 289-309. doi : 10.5802/jtnb.587. https://jtnb.centre-mersenne.org/item/JTNB_2007__19_1_289_0/

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