On the resolution of relative Thue equations

Gaál, István; Pohst, Michael

doi:10.1090/S0025-5718-01-01329-1

On the resolution of relative Thue equations
HTML articles powered by AMS MathViewer

by István Gaál and Michael Pohst;

Math. Comp. 71 (2002), 429-440

DOI: https://doi.org/10.1090/S0025-5718-01-01329-1

Published electronically: June 29, 2001

PDF | Request permission

Abstract:

An efficient algorithm is given for the resolution of relative Thue equations. The essential improvement is the application of an appropriate version of Wildanger’s enumeration procedure based on the ellipsoid method of Fincke and Pohst.

Recently relative Thue equations have gained an important application, e.g., in computing power integral bases in algebraic number fields. The presented methods can surely be used to speed up those algorithms.

The method is illustrated by numerical examples.

References

Alan Baker, Transcendental number theory, Cambridge University Press, London-New York, 1975. MR 422171
A. Baker and H. Davenport, The equations $3x^{2}-2=y^{2}$ and $8x^{2}-7=z^{2}$, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137. MR 248079, DOI 10.1093/qmath/20.1.129
A. Baker and G. Wüstholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19–62. MR 1234835, DOI 10.1515/crll.1993.442.19
Yuri Bilu and Guillaume Hanrot, Solving Thue equations of high degree, J. Number Theory 60 (1996), no. 2, 373–392. MR 1412969, DOI 10.1006/jnth.1996.0129
Yuri Bilu and Guillaume Hanrot, Thue equations with composite fields, Acta Arith. 88 (1999), no. 4, 311–326. MR 1690372, DOI 10.4064/aa-88-4-311-326
Yann Bugeaud and Kálmán Győry, Bounds for the solutions of Thue-Mahler equations and norm form equations, Acta Arith. 74 (1996), no. 3, 273–292. MR 1373714, DOI 10.4064/aa-74-3-273-292
M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger, KANT V4, J. Symbolic Comput. 24 (1997), no. 3-4, 267–283. Computational algebra and number theory (London, 1993). MR 1484479, DOI 10.1006/jsco.1996.0126
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
István Gaál, Computing all power integral bases in orders of totally real cyclic sextic number fields, Math. Comp. 65 (1996), no. 214, 801–822. MR 1333313, DOI 10.1090/S0025-5718-96-00708-9
István Gaál, Computing elements of given index in totally complex cyclic sextic fields, J. Symbolic Comput. 20 (1995), no. 1, 61–69. MR 1374224, DOI 10.1006/jsco.1995.1038
István Gaál, Application of Thue equations to computing power integral bases in algebraic number fields, Algorithmic number theory (Talence, 1996) Lecture Notes in Comput. Sci., vol. 1122, Springer, Berlin, 1996, pp. 151–155. MR 1446508, DOI 10.1007/3-540-61581-4_{5}1
I.Gaál, Solving index form equations in fields of degree nine with cubic subfields, J. Symbolic Comp. 30 (2000), 181–193.
István Gaál and Michael Pohst, On the resolution of index form equations in sextic fields with an imaginary quadratic subfield, J. Symbolic Comput. 22 (1996), no. 4, 425–434. MR 1428835, DOI 10.1006/jsco.1996.0060
Attila Pethő, On the resolution of Thue inequalities, J. Symbolic Comput. 4 (1987), no. 1, 103–109. MR 908418, DOI 10.1016/S0747-7171(87)80059-7
A. Pethő and R. Schulenberg, Effektives Lösen von Thue Gleichungen, Publ. Math. Debrecen 34 (1987), no. 3-4, 189–196 (German). MR 934900, DOI 10.5486/pmd.1987.34.3-4.04
Michael E. Pohst, Computational algebraic number theory, DMV Seminar, vol. 21, Birkhäuser Verlag, Basel, 1993. MR 1243639, DOI 10.1007/978-3-0348-8589-8
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, Thue and Thue-Mahler equations over rings of integers, J. London Math. Soc. (2) 56 (1997), no. 3, 455–462. MR 1610439, DOI 10.1112/S0024610797005619
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
B. M. M. de Weger, A Thue equation with quadratic integers as variables, Math. Comp. 64 (1995), no. 210, 855–861. MR 1277766, DOI 10.1090/S0025-5718-1995-1277766-0
K. Wildanger, Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern mit einer Anwendung auf die Bestimmung aller ganzen Punkte einer Mordellschen Kurve, Dissertation, Technical University, Berlin, 1997.