|
Parallel LLL-reduction for bounding the integral solutions of elliptic Diophantine equations
Author(s):
L.
Hajdu;
T.
Kovács.
Journal:
Math. Comp.
MSC (2000):
Primary 11G05;
Secondary 11Y50
Posted:
July 1, 2008
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
Stroeker and Tzanakis gave convincing numerical and heuristic evidence for the fact that in their  method a certain parameter  plays a decisive role in the size of the final bound for the integral points on elliptic curves. Furthermore, they provided an algorithm to determine the Mordell-Weil basis of the curve which corresponds to the optimal choice of  . In this paper we show that working with more Mordell-Weil bases simultaneously, the final bound for the integral points can be further decreased.
References:
-
- 1.
- A. Baker, The Diophantine equation
 , J. London Math. Soc. 43 (1968), 1-9. MR 0231783 (38:111) - 2.
- A. I. Barvinok, A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, in 34th Annual Symposium of Foundations of Computer Science, pp. 566-572, IEEE, Nov. 1993. MR 1328451
- 3.
- Y. Bugeaud, On the size of integer solutions of elliptic equations, Bull. Austral. Math. Soc. 57 (1998), 199-206. MR 1617363 (99h:11027)
- 4.
- Y. Bugeaud, On the size of integer solutions of elliptic equations II, Bull. Greek Math. Soc. 43 (2000), 125-130. MR 1846953 (2002e:11030)
- 5.
- S. David, Minorations de formes linéaires de logarithmes elliptiques, Soc. Math. France, Mémoire 62 (Suppl. Bull. S. M. F.) 123 (1995), pp. 143. MR 1385175 (98f:11078)
- 6.
- J. Gebel, A. Pethő, H. G. Zimmer, Computing integral points on elliptic curves, Acta Arith. 68 (1994), 171-192. MR 1305199 (95i:11020)
- 7.
- L. Hajdu, Optimal systems of fundamental
 -units for LLL-reduction (to appear). - 8.
- L. Hajdu, T. Herendi, Explicit bounds for the solutions of elliptic equations with rational coefficients, J. Symbolic Computation 25 (1998), 361-366. MR 1615334 (99a:11033)
- 9.
- L. Hajdu, T. Kovács, A parallel LLL-reduction method for bounding the solutions of
 -unit equations (manuscript). - 10.
- T. Kovács, Combinatorial Diophantine equations--the genus 1 case, Publ. Math. Debrecen 72 (2008), no. 1-2, 243-255. MR 2376872
- 11.
- J. A. De Loera, D. Haws, R. Hemmecke, P. Huggins, J. Tauzer, R. Yoshida, A user's guide for LattE v1.1, Nov. 2003.
- 12.
- Á. Pintér, On the magnitude of integer points on elliptic curves, Bull. Austral. Math. Soc. 52 (1995), 195-199. MR 1348477 (96k:11070)
- 13.
- D. Poulakis, Integral points on algebraic curves with exceptional units, J. Austral. Math. Soc. Ser. A 63 (1997), 145-164. MR 1475559 (98k:11088)
- 14.
- W. M. Schmidt, Integer points on curves of genus 1, Compositio Math. 81 (1992), 33-59. MR 1145607 (93e:11076)
- 15.
- T. N. Shorey, R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, Cambridge, 1986. MR 891406 (88h:11002)
- 16.
- V. G. Sprindžuk, Classical Diophantine Equations, Lecture Notes in Math. 1559, Springer-Verlag, Berlin, 1993. MR 1288309 (95g:11017)
- 17.
- R. J. Stroeker, N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms, Acta Arith. 67 (1994), 177-196. MR 1291875 (95m:11056)
- 18.
- R. J. Stroeker, N. Tzanakis, On the elliptic logarithm method for elliptic Diophantine equations: Reflections and an improvement, Experimental Math. 8 (1999), 135-149. MR 1700575 (2000d:11043)
- 19.
- R. J. Stroeker, N. Tzanakis, Computing all integer solutions of a genus 1 equation, Math. Comp. 72 (2003), 1935-1946. MR 1986812 (2004b:11037)
- 20.
- R. J. Stroeker, B. M. M. de Weger, Elliptic binomial Diophantine equations, Math. Comp. 68 (1999), 1257-1281. MR 1622097 (99i:11122)
- 21.
- N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations, Acta Arith. 75 (1996), 165-190. MR 1379397 (96m:11019)
- 22.
- P. M. Voutier, An upper bound for the size of integer solutions to
 , J. Number Theory 53 (1995), 247-271. MR 1348763 (96f:11049) - 23.
- B. M. M. de Weger, Algorithms for Diophantine equations, CWI Tract 65, Stichting Mathematisch Centrum, Amsterdam, 1989. MR 1026936 (90m:11205)
Similar Articles:
Retrieve articles in Mathematics of Computation
with MSC
(2000):
11G05,
11Y50
Retrieve articles in all Journals with MSC
(2000):
11G05,
11Y50
Additional Information:
L.
Hajdu
Affiliation:
University of Debrecen, Institute of Mathematics, and the Number Theory Research Group of the Hungarian Academy of Sciences, P.O. Box 12, H-4010 Debrecen, Hungary
Email:
hajdul@math.klte.hu
T.
Kovács
Affiliation:
University of Debrecen, Institute of Mathematics, P.O. Box 12, H-4010 Debrecen, Hungary
Email:
tundekov@gmail.com
DOI:
10.1090/S0025-5718-08-02160-1
PII:
S 0025-5718(08)02160-1
Keywords:
Elliptic curves,
integral points,
LLL-reduction
Received by editor(s):
December 18, 2007
Received by editor(s) in revised form:
March 12, 2008
Posted:
July 1, 2008
Additional Notes:
Research supported in part by the Hungarian Academy of Sciences and by the OTKA grants T48791 and K67580.
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org