Effective solution of families of Thue equations containing several parameters

Abstract

F (X,Y ) = m, where F ∈ Z[X,Y ] is an irreducible form of degree n ≥ 3 and m 6= 0 a fixed integer, has only finitely many solutions. However, this proof is non-effective and does not give any bounds for the size of the possible solutions. In 1968, A. Baker could give effective bounds based on his famous theory on linear forms in logarithms of algebraic numbers. In the last decades, this method was refined (see for instance Baker and Wustholz [1] and Waldschmidt [29]). Recent explicit upper bounds for the solutions of Thue equations have been given by Bugeaud and Győry [3]. Algorithms for the solution of a single Thue equation have been developed by several authors (see Bilu and Hanrot [2]). E. Thomas [25] was the first to deal with a parametrized family of Thue equations; since then, some families have been investigated: cubic families have been discussed by Mignotte [16], Lee [12], and Mignotte and Tzanakis [19], a cubic inequality has been solved by Mignotte, Pethő, and Lemmermeyer [17]; quartic families have been considered by Pethő [20], Mignotte, Pethő, and Roth [18], Lettl and Pethő [13], Chen and Voutier [4], and Heuberger, Pethő, and Tichy [10]; Wakabayashi [28] dealt with a quartic inequality, Pethő and Tichy [21] solved a two-parametric quartic family, quintic families have been investigated by Heuberger [9] and Gaal and Lettl [6], a sextic family has been solved by Lettl, Pethő, and Voutier [14, 15]; see