Abstract
In this paper, we provide first a new algorithm for testing whether a monomial ideal is in Nœther position or not, without using its dimension, within a complexity which is quadratic in input size. Using this algorithm, we provide also a new algorithm to put an ideal in this position within an incremental (one variable after the other) random linear change of the last variables without using its dimension. We describe a modular (probabilistic) version of these algorithms for any ideal using the modular method used in [2] with some modifications. These algorithms have been implemented in the distributed library noether.lib [17] of Singular, and we evaluate their performance via some examples.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arnold, E.A.: Computing Gröbner Bases with Hilbert Lucky Primes. PhD thesis, University of Maryland (2000)
Arnold, E.A.: Modular algorithms for computing Gröbner bases. J. Symbolic Comput. 35(4), 403–419 (2003)
Bardet, M.: Étude des systèmes algébriques surdéterminés: Applications aux codes correcteurs et à la cryptographie. PhD thesis, Université Paris6 (2004)
Bayer, D., Stillman, M.: Computation of Hilbert functions. J. Symbolic Comput. 14(1), 31–50 (1992)
Bermejo, I., Gimenez, Ph., Greuel, G.-M.: mregular.lib. A Singular 3.0.1 distributed library for computing the Castelnuovo-Mumford regularity (2005)
Bermejo, I., Gimenez, P.: Computing the Castelnuovo-Mumford regularity of some subschemes of \({\mathbb P}\sb K\sp n\) using quotients of monomial ideals. J. Pure Appl. Algebra 164(1-2), 23–33 (2001); Effective methods in algebraic geometry (Bath, 2000)
Bermejo, I., Gimenez, P.: Saturation and Catelnuovo-Mumford regularity. J. Algebra 303, 592–617 (2006)
Bigatti, A.M., Conti, P., Robbiano, L., Traverso, C.: A Divide and Conquer algorithm for Hilbert-Poincaré series, multiplicity and dimension of monomial ideals. In: Moreno, O., Cohen, G., Mora, T. (eds.) AAECC 1993. LNCS, vol. 673, pp. 76–88. Springer, Heidelberg (1993)
Cox, D., Little, J., O’Shea, D.: Ideals, varieties, and algorithms. In: Undergraduate Texts in Mathematics, 2nd edn. Springer, New York (1997); (An introduction to computational algebraic geometry and commutative algebra)
Decker, W., Greuel, G.-M., Pfister, G.: Primary decomposition: algorithms and comparisons. In: Algorithmic algebra and number theory (Heidelberg, 1997), pp. 187–220. Springer, Berlin (1999)
Eisenbud, D.: Commutative algebra with a view toward algebraic geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)
Fröberg, R.: An introduction to Gröbner bases, New York. Pure and Applied Mathematics. John Wiley & Sons Ltd., Chichester (1997)
Giusti, M., Hägele, K., Lecerf, G., Marchand, J., Salvy, B.: The projective Noether Maple package: computing the dimension of a projective variety. J. Symbolic Comput. 30(3), 291–307 (2000)
Grayson, D.R., Stillman, M.E.: Macaulay 2 version 1.1. software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/
Greuel, G.-M., Pfister, G.: A Singular introduction to commutative algebra. Springer, Berlin (2002); With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann, With 1 CD-ROM (Windows, Macintosh, and UNIX)
Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3.0.3. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern (2005), http://www.singular.uni-kl.de
Hashemi, A.: noether.lib. A Singular 3.0.3 distributed library for computing the nœther normalization (2007)
Lecerf, G.: Computing the Equidimensional Decomposition of an Algebraic Closed Set by means of Lifting Fibers. Journal of Complexity 19(4), 564–596 (2003)
Krick, T., Logar, A.: An algorithm for the computation of the radical of an ideal in the ring of polynomials. In: Mattson, H.F., Rao, T.R.N., Mora, T. (eds.) AAECC 1991. LNCS, vol. 539, pp. 195–205. Springer, Heidelberg (1991)
Logar, A.: A Computational Proof of the Noether Normalization Lemma. In: Mora, T. (ed.) AAECC 1988. LNCS, vol. 357, pp. 259–273. Springer, Heidelberg (1989)
Pauer, F.: On lucky ideals for Gröbner basis computations. J. Symbolic Comput. 14, 471–482 (1992)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hashemi, A. (2008). Efficient Algorithms for Computing Nœther Normalization. In: Kapur, D. (eds) Computer Mathematics. ASCM 2007. Lecture Notes in Computer Science(), vol 5081. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87827-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-87827-8_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87826-1
Online ISBN: 978-3-540-87827-8
eBook Packages: Computer ScienceComputer Science (R0)