Solving superelliptic Diophantine equations by Baker's method |
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Authors: | Yuri F Bilu Guillaume Hanrot |
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Institution: | (1) Mathematisches Institut, Universitä at Basel, Rheinsprung 21, CH-4051 Basel, Switzerland;(2) Algorithmique Expérimentale (A2X), UMR CNRS 9936, Université Bordeaux 1, 351, cours de la Libération, F-33405 Talence Cedex, France |
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Abstract: | We describe a method for complete solution of the superelliptic Diophantine equation ayp=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 3] 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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Keywords: | Diophantine equations linear forms in logarithms |
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