Solving superelliptic diophantine equations by Baker's method

Yuri Bilu , Guillaume Hanrot (1)

1 POLKA - Polynomials, Combinatorics, Arithmetic

We describe a method for complete solution of the superelliptic Diophantine equation ay^p=f(x). The method is based on Baker's theory of linear forms in the logarithms. The characteristic feature of our approach (as compared with the classical method is that we reduce the equation directly to the linear forms in logarithms, without intermediate use of Thue and linear unit equations. We show that the reduction method of Baker and Davenport is applicable for superelliptic equations, and develop a very efficient method for enumerating the solutions below the reduced bound. The method requires computing the algebraic data in number fields of degree pn(n-1)/2 at most; in many cases this number can be reduced. Two examples with p=3 and n=4 are given.

Mots clés: soution of diophantine equations; méthode de Baker; Baker's method; résolution d'équations diophantiennes

Domaines: Autre [cs.OH]

Yuri Bilu, Guillaume Hanrot. Solving superelliptic diophantine equations by Baker's method. Compositio Mathematica, 1998, 112 (3), pp.273--312. ⟨10.1023/A:1000305028888⟩. ⟨inria-00098523⟩