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

Let $F\in \mathbb{Z}[x,y]$ be an irreducible binary form of degree $d\ge 7$ and content one. Let $\alpha$ be a root of $F(x,1)$ and assume that the field extension $\mathbb{Q}(\alpha )\mathrm{/}\mathbb{Q}$ is Galois. We prove that, for every sufficiently large prime power $p^k$, the number of solutions to the Diophantine equation of Thue type $$\mathrm{\mid }F(x,y)\mathrm{\mid }=t{p}^k$$ in integers $(x,y,t)$ such that $\gcd (x,y)=1$ and $1\le t\le (p^k{)}^{\lambda }$ 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 $$\mathrm{\mid }F(x,y)\mathrm{\mid }=t{p}^z$$ in integers $(x,y,z,t)$ such that $\gcd (x,y)=1$, $z\ge 1$ and $1\le t\le (p^z{)}^{\frac{10d-61}{20d+40}}$ 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.