Determining the small solutions to 𝑆-unit equations

Smart, N.

doi:10.1090/S0025-5718-99-01140-0

Determining the small solutions to $S$-unit equations
HTML articles powered by AMS MathViewer

by N. P. Smart PDF

Math. Comp. 68 (1999), 1687-1699 Request permission

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

Johannes Buchmann, Michael J. Jacobson Jr., and Edlyn Teske, On some computational problems in finite abelian groups, Math. Comp. 66 (1997), no. 220, 1663–1687. MR 1432126, DOI 10.1090/S0025-5718-97-00880-6
U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comp. 44 (1985), no. 170, 463–471. MR 777278, DOI 10.1090/S0025-5718-1985-0777278-8
I. M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), 590–622. MR 81, DOI 10.1215/S0012-7094-39-00549-1
N. P. Smart, The solution of triangularly connected decomposable form equations, Math. Comp. 64 (1995), no. 210, 819–840. MR 1277771, DOI 10.1090/S0025-5718-1995-1277771-4
N. P. Smart, $S$-unit equations, binary forms and curves of genus $2$, Proc. London Math. Soc. (3) 75 (1997), no. 2, 271–307. MR 1455857, DOI 10.1112/S002461159700035X
Edlyn Teske, A space efficient algorithm for group structure computation, Math. Comp. 67 (1998), no. 224, 1637–1663. MR 1474658, DOI 10.1090/S0025-5718-98-00968-5
N. Tzanakis and B. M. M. de Weger, How to explicitly solve a Thue-Mahler equation, Compositio Math. 84 (1992), no. 3, 223–288. MR 1189890
B. M. M. de Weger, Solving exponential Diophantine equations using lattice basis reduction algorithms, J. Number Theory 26 (1987), no. 3, 325–367. MR 901244, DOI 10.1016/0022-314X(87)90088-6
B. M. M. de Weger, Algorithms for Diophantine equations, CWI Tract, vol. 65, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. MR 1026936
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.