Abstract
We discuss the problem of determining reduction numbers of a polynomial ideal \(\mathcal{I}\) in n variables. We present two algorithms based on parametric computations. The first one determines the absolute reduction number of \(\mathcal{I}\) and requires computations in a polynomial ring with (n–dim\(\mathcal{I}\))dim\(\mathcal{I}\) parameters and n–dim\(\mathcal{I}\) variables. The second one computes via a Gröbner system the set of all reduction numbers of the ideal \(\mathcal{I}\) and thus in particular also its big reduction number. However, it requires computations in a ring with ndim\(\mathcal{I}\) parameters and n variables.
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
Albert, M., Hashemi, A., Pytlik, P., Schweinfurter, M., Seiler, W.: Effective genericity for polynomial ideals (in preparation, 2014)
Bayer, D., Stillman, M.: A theorem on refining division orders by the reverse lexicographic orders. Duke J. Math. 55, 321–328 (1987)
Bresinsky, H., Hoa, L.: On the reduction number of some graded algebras. Proc. Amer. Math. Soc. 127, 1257–1263 (1999)
Conca, A.: Reduction numbers and initial ideals. Proc. Amer. Math. Soc. 131, 1015–1020 (2002)
Galligo, A.: A propos du théorème de préparation de Weierstrass. In: Norguet, F. (ed.) Fonctions de Plusieurs Variables Complexes. Lecture Notes in Mathematics, vol. 409, pp. 543–579. Springer, Berlin (1974)
Galligo, A.: Théorème de division et stabilité en géometrie analytique locale. Ann. Inst. Fourier 29(2), 107–184 (1979)
Green, M.: Generic initial ideals. In: Elias, J., Giral, J., Miró-Roig, R., Zarzuela, S. (eds.) Six Lectures on Commutative Algebra. Progress in Mathematics, vol. 166, pp. 119–186. Birkhäuser, Basel (1998)
Greuel, G.M., Pfister, G.: A Singular Introduction to Commutative Algebra, 2nd edn. Springer, Berlin (2008)
Hashemi, A., Schweinfurter, M., Seiler, W.: Quasi-stability versus genericity. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2012. LNCS, vol. 7442, pp. 172–184. Springer, Heidelberg (2012)
Kapur, D., Sun, Y., Wang, D.: A new algorithm for computing comprehensive Gröbner systems. In: Koepf, W. (ed.) Proc. ISSAC 2010, pp. 29–36. ACM Press (2010)
Kapur, D., Sun, Y., Wang, D.: An efficient algorithm for computing a comprehensive Gröbner system of a parametric polynomial system. J. Symb. Comput. 49, 27–44 (2013)
Mayr, E., Meyer, A.: The complexity of the word problems for commutative semigroups and polynomial ideals. Adv. Math. 46, 305–329 (1982)
Montes, A.: A new algorithm for discussing Gröbner bases with parameters. J. Symb. Comput. 33, 183–208 (2002)
Montes, A., Wibmer, M.: Gröbner bases for polynomial systems with parameters. J. Symb. Comput. 45, 1391–1425 (2010)
Northcott, D., Rees, D.: Reduction of ideals in local rings. Cambridge Philos. Soc. 50, 145–158 (1954)
Sato, Y., Suzuki, A.: A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases. In: Dumas, J.-G (ed.) Proc. ISSAC 2006, pp. 326–331. ACM Press (2006)
Seiler, W.: A combinatorial approach to involution and δ-regularity I: Involutive bases in polynomial algebras of solvable type. Appl. Alg. Eng. Comm. Comp. 20, 207–259 (2009)
Seiler, W.: A combinatorial approach to involution and δ-regularity II: Structure analysis of polynomial modules with Pommaret bases. Appl. Alg. Eng. Comm. Comp. 20, 261–338 (2009)
Trung, N.: Reduction exponent and degree bounds for the defining equations of a graded ring. Proc. Amer. Math. Soc. 101, 229–236 (1987)
Trung, N.: Gröbner bases, local cohomology and reduction number. Proc. Amer. Math. Soc. 129, 9–18 (2001)
Trung, N.: Constructive characterization of the reduction numbers. Compos. Math. 137, 99–113 (2003)
Vasconcelos, W.: The reduction number of an algebra. Compos. Math. 106, 189–197 (1996)
Vasconcelos, W.: Reduction numbers of ideals. J. Alg. 216, 652–664 (1999)
Weispfenning, V.: Comprehensive Gröbner bases. J. Symb. Comp. 14, 1–29 (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Hashemi, A., Schweinfurter, M., Seiler, W.M. (2014). Deterministically Computing Reduction Numbers of Polynomial Ideals. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2014. Lecture Notes in Computer Science, vol 8660. Springer, Cham. https://doi.org/10.1007/978-3-319-10515-4_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-10515-4_14
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10514-7
Online ISBN: 978-3-319-10515-4
eBook Packages: Computer ScienceComputer Science (R0)