Absolute Bound On the Number of Solutions of Certain Diophantine Equations of Thue and Thue–Mahler Type

Mosunov, Anton

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 947-965

Let $F\in \mathbb{Z}[x,y]$ be an irreducible binary form of degree $d\ge 7$ and content one. Let $\alpha$ be a complex root of $F(x,1)$ and assume that the field extension $\mathbb{Q}\left(\alpha \right)/\mathbb{Q}$ is Galois. We prove that, for every sufficiently large prime power $p^k$, the number of solutions to the Diophantine equation of Thue type

$$\left|F(x,y)\right|=h{p}^k$$

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

$$gcd(x,y)=1\text{and}1\le h\le {\left(p^k\right)}^{\lambda }$$

does not exceed $24$. Here $\lambda =\lambda \left(d\right)$ 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

$$\left|F(x,y)\right|=h{p}^z$$

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

$$gcd(x,y)=1,z\ge 1\text{and}1\le h\le {\left(p^z\right)}^{\frac{10d-61}{20d+40}}$$

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.

Soit $F$ une forme binaire irréductible de degré $d\ge 7$ à coefficients entiers dont le plus grand diviseur commun est égal à $1.$ Soit $\alpha$ une racine complexe de $F(x,1).$ Supposons que l’extension $\mathbb{Q}\left(\alpha \right)/\mathbb{Q}$ est galoisienne. Nous prouvons que, pour toute puissance $p^k$ suffisamment grande d’un nombre premier $p$, le nombre de solutions de l’équation diophantienne de type Thue

$$\left|F(x,y)\right|=h{p}^k$$

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

$$gcd(x,y)=1\text{et}1\le h\le {\left(p^k\right)}^{\lambda }$$

est borné par $24$. Ici $\lambda =\lambda \left(d\right)$ 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

$$\left|F(x,y)\right|=h{p}^z$$

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

$$gcd(x,y)=1,z\ge 1\text{et}1\le h\le {\left(p^z\right)}^{\frac{10d-61}{20d+40}}$$

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.

Classification : 11D59
Keywords: Thue equation, Thue–Mahler equation, Diophantine approximation, binary form

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