Absolute Bound On the Number of Solutions of Certain Diophantine Equations of Thue and Thue–Mahler Type
Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 947-965.

Soit F une forme binaire irréductible de degré d7 à coefficients entiers dont le plus grand diviseur commun est égal à 1. Soit α une racine complexe de F(x,1). Supposons que l’extension (α)/ est galoisienne. Nous prouvons que, pour toute puissance pk suffisamment grande d’un nombre premier p, le nombre de solutions de l’équation diophantienne de type Thue

|F(x,y)|=hpk

en nombres entiers (x,y,h) tels que

gcd(x,y)=1et1h(pk)λ

est borné par 24. Ici λ=λ(d) est une fonction positive et monotone croissante qui s’approche de 1 lorsque d tend vers l’infini. Nous prouvons également que, pour tout nombre premier p suffisamment grand, le nombre de solutions de l’équation diophantienne de type Thue–Mahler

|F(x,y)|=hpz

en entiers (x,y,z,h) tels que

gcd(x,y)=1,z1et1h(pz)10d-6120d+40

ne dépasse pas 3984. Nos preuves découlent de la combinaison de deux principes d’approximation diophantienne, à savoir le principe d’écart non-archimédien généralisé et le principe de Thue–Siegel.

Let F[x,y] be an irreducible binary form of degree d7 and content one. Let α be a complex root of F(x,1) and assume that the field extension (α)/ is Galois. We prove that, for every sufficiently large prime power pk, the number of solutions to the Diophantine equation of Thue type

|F(x,y)|=hpk

in integers (x,y,h) such that

gcd(x,y)=1and1h(pk)λ

does not exceed 24. Here λ=λ(d) is a certain positive, monotonously increasing function that approaches one as d tends to infinity. We also prove that, for every sufficiently large prime number p, the number of solutions to the Diophantine equation of Thue–Mahler type

|F(x,y)|=hpz

in integers (x,y,z,h) such that

gcd(x,y)=1,z1and1h(pz)10d-6120d+40

does not exceed 3984. Our proofs follow from the combination of two principles of Diophantine approximation, namely the generalized non-Archimedean gap principle and the Thue–Siegel principle.

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DOI : 10.5802/jtnb.1301
Classification : 11D59
Mots-clés : Thue equation, Thue–Mahler equation, Diophantine approximation, binary form

Anton Mosunov 1

1 Cornell University 212 Garden Avenue Ithaca, NY 14853, USA
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Anton Mosunov. Absolute Bound On the Number of Solutions of Certain Diophantine Equations of Thue and Thue–Mahler Type. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 947-965. doi : 10.5802/jtnb.1301. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1301/

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