Solving superelliptic diophantine equations by Baker's method

Yuri Bilu Guillaume Hanrot 1
1 POLKA - Polynomials, Combinatorics, Arithmetic
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : 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.
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Compositio Mathematica, Foundation Compositio Mathematica, 1998, 112 (3), pp.273--312. 〈10.1023/A:1000305028888〉
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Soumis le : lundi 25 septembre 2006 - 17:03:06
Dernière modification le : jeudi 11 janvier 2018 - 06:19:48

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

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