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

Let F ∈ ℤ [ x , y ] be an irreducible binary form of degree d ≥ 7 and content one. Let α be a complex root of F ( x , 1 ) and assume that the field extension ℚ ( α ) / 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 ) | = h p k in integers ( x , y , h ) such that gcd ( x , y ) = 1 and 1 ≤ h ≤ ( p k ) 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 ) | = h p z in integers ( x , y , z , h ) such that gcd ( x , y ) = 1 , z ≥ 1 and 1 ≤ h ≤ ( p z ) 10 d - 61 20 d + 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.