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

TL;DR

This paper establishes explicit upper bounds on the number of solutions to certain Thue and Thue-Mahler equations involving large prime powers, using advanced Diophantine approximation techniques.

Contribution

It provides the first explicit bounds on solutions for these equations when the degree is at least 7, extending previous qualitative results.

Findings

Bound of 24 solutions for Thue equations with large prime powers

Bound of 1992 solutions for Thue-Mahler equations with large primes

Solutions are limited under specified conditions involving gcd and size constraints

Abstract

Let $F \in \mathbb Z[x, y]$ be an irreducible binary form of degree $d \geq 7$ and content one. Let $\alpha$ be a root of $F(x, 1)$ and assume that the field extension $\mathbb Q(\alpha)/\mathbb Q$ is Galois. We prove that, for every sufficiently large prime power $p^k$, the number of solutions to the Diophantine equation of Thue type $$ |F(x, y)| = tp^k $$ in integers $(x, y, t)$ such that $\gcd(x, y) = 1$ and $1 \leq t \leq (p^k)^\lambda$ does not exceed $24$. Here $\lambda = \lambda(d)$ 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 $$ |F(x, y)| = tp^z $$ in integers $(x, y, z, t)$ such that $\gcd(x, y) = 1$, $z \geq 1$ and $1 \leq t \leq (p^z)^{\frac{10d - 61}{20d + 40}}$ does not exceed 1992. Our proofs follow from the combination of two principles of Diophantine approximation, namely the generalized non-Archimedean gap principle and the Thue-Siegel principle.