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Hermite and Smith normal form algorithms over Dedekind domains
Author(s):
Henri
Cohen.
Journal:
Math. Comp.
65
(1996),
1681-1699.
MSC (1991):
Primary 11Y40
MathSciNet review:
1361805
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Abstract:
We show how the usual algorithms valid over Euclidean domains, such as the Hermite Normal Form, the modular Hermite Normal Form and the Smith Normal Form can be extended to Dedekind rings. In a sequel to this paper, we will explain the use of these algorithms for computing in relative extensions of number fields.
References:
- 1.
- W. Bosma and M. Pohst, Computations with finitely generated modules over Dedekind rings, Proceedings ISSAC'91 (1991), 151--156.
- 2.
- H. Cohen, A course in computational algebraic number theory, Graduate Texts in Math., vol. 138, Springer-Verlag, 1993. MR 94i:11105
- 3.
- H. Cohen, F. Diaz y Diaz and M. Olivier, Algorithmic computations in relative extensions of number fields, in preparation.
- 4.
- P. D. Domich, R. Kannan and L. E. Trotter, Jr., Hermite normal form computation using modulo determinant arithmetic, Math. Oper. Research 12 (1987), 50--59. MR 88e:65047
- 5.
- J. Hafner and K. McCurley, Asymptotically fast triangularization of matrices over rings, SIAM J. Comput. 20 (1991), 1068--1083. MR 93d:15021
- 6.
- G. Havas and B. Majewski, Hermite normal form computation for integer matrices, Congr. Numer. 105 (1994), 184--193. CMP 96:10
- 7.
- R. Kannan and A. Bachem, Polynomial algorithms for computing the Smith and Hermite normal form of an integer matrix, SIAM J. Comput. 8 (1979), 499-507. MR 81k:15002
- 8.
- P. Montgomery, in preparation.
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Additional Information:
Henri
Cohen
Affiliation:
Laboratoire A2X, UMR 9936 du C.N.R.S., Université Bordeaux I, 351 Cours de la Libération, 33405 Talence Cedex, France
Email:
cohen@math.u-bordeaux.fr
DOI:
10.1090/S0025-5718-96-00766-1
PII:
S 0025-5718(96)00766-1
Keywords:
Dedekind domain,
Hermite normal form,
Smith normal form,
relative extensions of number fields
Received by editor(s):
January 11, 1995
Received by editor(s) in revised form:
July 19, 1995
Copyright of article:
Copyright
1996,
American Mathematical Society
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