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Determining the small solutions
to $S$-unit equations

Author: N. P. Smart
Journal: Math. Comp. 68 (1999), 1687-1699
MSC (1991): Primary 11Y50, 11D61
Published electronically: March 11, 1999
MathSciNet review: 1653990
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Abstract: In this paper we generalize the method of Wildanger for finding small solutions to unit equations to the case of $S$-unit equations. The method uses a minor generalization of the LLL based techniques used to reduce the bounds derived from transcendence theory, followed by an enumeration strategy based on the Fincke-Pohst algorithm. The method used reduces the computing time needed from MIPS years down to minutes.

References [Enhancements On Off] (What's this?)

  • 1. J. Buchmann, M. Jacobson, and E. Teske. On some computational problems in finite abelian groups. Math. Comp., 66:1663-1687, 1997. MR 98a:11185
  • 2. U. Fincke and M. Pohst. Improved methods for calculating vectors of short length in a lattice, including a complexity analysis. Math. Comp., 44:463-471, 1985. MR 86e:11050
  • 3. K. Gy\H{o}ry. On the number of solutions of linear equations in units of an algebraic number field. Comment. Math. Helvetici, 54:585-600, 1979. MR 81g:11031
  • 4. N.P. Smart. The solution of triangularly connected decomposable form equations. Math. Comp., 64:819-840, 1995. MR 95f:11110
  • 5. N.P. Smart. S-unit equations, binary forms and curves of genus 2. Proc. London Math. Soc., 75:271-307, 1997. MR 98d:11072
  • 6. E. Teske. A space efficient algorithm for group structure computation. Math. Comp., 67:1637-1663, 1998. MR 99a:11146
  • 7. N. Tzanakis and B.M.M. de Weger. How to explicitly solve a Thue-Mahler equation. Compositio Math., 84:223-288, 1992; 89 (1993), 241-242. MR 93k:11025; MR 95a:11030
  • 8. B.M.M. de Weger. Solving exponential diophantine equations using lattice basis reduction algorithms. J. Number Theory, 26:325-367, 1987; 31 (1989), 88-89. MR 88k:11097; MR 90a:11040
  • 9. B.M.M. de Weger. Algorithms for Diophantine Equations. Centre for Mathematics and Computer Science Amsterdam, 1989. CWI-Tract 65. MR 90m:11205
  • 10. K. Wildanger. Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern mit einer Anwendung auf die Bestimmung aller ganzen Punkte einer Mordellschen Kurve. PhD thesis, Technischen Universität Berlin, 1997.

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Additional Information

N. P. Smart
Affiliation: Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol, BS12 6QZ, U.K.

Keywords: $S$-unit equations
Received by editor(s): December 1, 1997
Published electronically: March 11, 1999
Article copyright: © Copyright 1999 American Mathematical Society

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