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Abstract:

Given a reduced affine algebra A over a perfect field K, we present parallel algorithms to compute the normalization Ā of A. Our starting point is the algorithm of Greuel et al. (2010), which is an improvement of de Jong's algorithm (de Jong, 1998; Decker et al., 1999). First, we propose to stratify the singular locus Sing(A) in a way which is compatible with normalization, apply a local version of the normalization algorithm at each stratum, and find Ā by putting the local results together. Second, in the case where K=Q is the field of rationals, we propose modular versions of the global and local-to-global algorithms. We have implemented our algorithms in the computer algebra system Singular and compare their performance with that of the algorithm of Greuel et al. (2010). In the case where K=Q, we also discuss the use of modular computations of Gröbner bases, radicals, and primary decompositions. We point out that in most examples, the new algorithms outperform the algorithm of Greuel et al. (2010) by far, even if we do not run them in parallel. © 2012 Elsevier B.V.

Registro:

Documento: Artículo
Título:Parallel algorithms for normalization
Autor:Böhm, J.; Decker, W.; Laplagne, S.; Pfister, G.; Steenpaß, A.; Steidel, S.
Filiación:Department of Mathematics, University of Kaiserslautern, Erwin-Schrödinger-Str., 67663 Kaiserslautern, Germany
Departamento de Matemática, FCEN, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, (C1428EGA) Buenos Aires, Argentina
Palabras clave:Grauert-Remmert criterion; Integral closure; Modular computation; Normalization; Parallel computation; Test ideal
Año:2013
Volumen:51
Página de inicio:99
Página de fin:114
DOI: http://dx.doi.org/10.1016/j.jsc.2012.07.002
Título revista:Journal of Symbolic Computation
Título revista abreviado:J. Symb. Comput.
ISSN:07477171
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_07477171_v51_n_p99_Bohm.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_07477171_v51_n_p99_Bohm

Referencias:

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  • Böhm, J., Decker, W., Laplagne, S., Seelisch, F., in preparation. Computing integral bases via localization and Hensel lifting; Böhm, J., Decker, W., Fieker, C., Pfister, G., The use of bad primes in rational reconstruction, , http://arxiv.org/abs/1207.1651, Preprint, available at
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  • Decker, W., Greuel, G.-M., Pfister, G., de Jong, T., The normalization: A new algorithm, implementation and comparisons (1999) Computational Methods for Representations of Groups and Algebras, , Birkhäuser
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Citas:

---------- APA ----------
Böhm, J., Decker, W., Laplagne, S., Pfister, G., Steenpaß, A. & Steidel, S. (2013) . Parallel algorithms for normalization. Journal of Symbolic Computation, 51, 99-114.
http://dx.doi.org/10.1016/j.jsc.2012.07.002
---------- CHICAGO ----------
Böhm, J., Decker, W., Laplagne, S., Pfister, G., Steenpaß, A., Steidel, S. "Parallel algorithms for normalization" . Journal of Symbolic Computation 51 (2013) : 99-114.
http://dx.doi.org/10.1016/j.jsc.2012.07.002
---------- MLA ----------
Böhm, J., Decker, W., Laplagne, S., Pfister, G., Steenpaß, A., Steidel, S. "Parallel algorithms for normalization" . Journal of Symbolic Computation, vol. 51, 2013, pp. 99-114.
http://dx.doi.org/10.1016/j.jsc.2012.07.002
---------- VANCOUVER ----------
Böhm, J., Decker, W., Laplagne, S., Pfister, G., Steenpaß, A., Steidel, S. Parallel algorithms for normalization. J. Symb. Comput. 2013;51:99-114.
http://dx.doi.org/10.1016/j.jsc.2012.07.002