Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T19:33:58.046Z Has data issue: false hasContentIssue false
  • English
  • Français

Solving superelliptic Diophantine equations by Baker‘s method

Published online by Cambridge University Press:  04 December 2007

YURI F. BILU  and
GUILLAUME HANROT
YURI F. BILU
Affiliation:
Mathematisches Institut, Universitä at Basel Rheinsprung 21, CH-4051 Basel, Switzerland; e-mail: yuri@math.unibas.ch
GUILLAUME HANROT
Affiliation:
Algorithmique Expérimentale (A2X), UMR CNRS 9936, Université Bordeaux 1, 351, cours de la Libération, F-33405 Talence Cedex France, e-mail: hanrot@math.u-bordeaux.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 [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.

Keywords

Diophantine equationslinear forms in logarithms.
Type
Research Article
Information
Compositio Mathematica , Volume 112 , Issue 3 , July 1998 , pp. 273 - 312
Copyright
© 1998 Kluwer Academic Publishers