Explicit bounds for the solutions of superelliptic equations over number fields

Abstract Let f be a polynomial with coefficients in the ring O S {O_{S}} of S -integers of a number field K , b a non-zero S -integer, and m an integer ≥ 2 {\geq 2} . We consider the following equation ( ⋆ ) {(\star)} : f ⁢ ( x ) = b ⁢ y m {f(x)=by^{m}} in x , y ∈ O S {x,y\in O_{S}} . Under the well-known LeVeque condition, we give fully explicit upper bounds in terms of K , S , f , m {K,S,f,m} and the S -norm of b for the heights of the solutions x of equation ( ⋆ ) {(\star)} . Further, we give an explicit bound C in terms of K , S , f {K,S,f} and the S -norm of b such that if m > C {m>C} equation ( ⋆ ) {(\star)} has only solutions with y = 0 {y=0} or a root of unity. Our results are more detailed versions of work of Treli