Mathematics of Computation

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ISSN 1088-6842(e) ISSN 0025-5718(p)

Efficient computation of Castelnuovo-Mumford regularity

Author: Amir Hashemi
Journal: Math. Comp. 81 (2012), 1163-1177
MSC (2010): Primary 14Q20, 13P10
Posted: July 21, 2011
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Abstract: In this paper, we introduce the notion of a homogeneous ideal in quasi stable position (QSP); a new definition for the notion of generic coordinates to compute efficiently the Castelnuovo-Mumford regularity of a homogeneous ideal. This definition is simple to check, because it is tested on the initial ideal for the degree reverse lexicographic ordering. It is explicit, because we provide an algorithm to decide whether a monomial ideal is in QSP or not. The main result of this paper is that the Castelnuovo-Mumford regularity of an ideal in QSP is the maximal degree of the elements of its reduced Gröbner basis with respect to the reverse lexicographic ordering. We have implemented an algorithm in (the distributed library noether.lib of) SINGULAR based on the above results for computing the Castelnuovo-Mumford regularity of a general ideal, and we evaluate its performance via some examples.

References

Bibliography

1. Amir Hashemi.
noether.lib.
A SINGULAR 3.0.3 distributed library for computing Castelnuovo-Mumford regularity, 2007.
2. Dave Bayer.
The Division Algorithm and the Hilbert Scheme.
PhD thesis, Harvard University, 1982.
3. David Bayer and Michael Stillman, A criterion for detecting 𝑚-regularity, Invent. Math. 87 (1987), no. 1, 1–11. MR 862710 (87k:13019), http://dx.doi.org/10.1007/BF01389151
4. Isabel Bermejo and Philippe Gimenez, Computing the Castelnuovo-Mumford regularity of some subschemes of ℙ_{𝕂}ⁿ using quotients of monomial ideals, J. Pure Appl. Algebra 164 (2001), no. 1-2, 23–33. Effective methods in algebraic geometry (Bath, 2000). MR 1854328 (2002i:13013), http://dx.doi.org/10.1016/S0022-4049(00)00143-2
5. Isabel Bermejo and Philippe Gimenez, Saturation and Castelnuovo-Mumford regularity, J. Algebra 303 (2006), no. 2, 592–617. MR 2255124 (2007f:13026), http://dx.doi.org/10.1016/j.jalgebra.2005.05.020
6. Wolfram Decker, Gert-Martin Greuel, and Gerhard Pfister, Primary decomposition: algorithms and comparisons, Algorithmic algebra and number theory (Heidelberg, 1997) Springer, Berlin, 1999, pp. 187–220. MR 1672046 (99m:13049)
7. David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960 (97a:13001)
8. David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89–133. MR 741934 (85f:13023), http://dx.doi.org/10.1016/0021-8693(84)90092-9
9. Shalom Eliahou and Michel Kervaire, Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), no. 1, 1–25. MR 1037391 (91b:13019), http://dx.doi.org/10.1016/0021-8693(90)90237-I
10. André Galligo, À propos du théorème de-préparation de Weierstrass, Fonctions de plusieurs variables complexes (Sém. François Norguet, octobre 1970–décembre 1973; à la mémoire d’André Martineau), Springer, Berlin, 1974, pp. 543–579. Lecture Notes in Math., Vol. 409 (French). Thèse de 3ème cycle soutenue le 16 mai 1973 à l’Institut de Mathématique et Sciences Physiques de l’Université de Nice. MR 0402102 (53 #5924)
11. M. Giusti, Some effectivity problems in polynomial ideal theory, EUROSAM 84 (Cambridge, 1984) Lecture Notes in Comput. Sci., vol. 174, Springer, Berlin, 1984, pp. 159–171. MR 779123 (86d:12001), http://dx.doi.org/10.1007/BFb0032839
12. Mark L. Green, Generic initial ideals, Six lectures on commutative algebra (Bellaterra, 1996) Progr. Math., vol. 166, Birkhäuser, Basel, 1998, pp. 119–186. MR 1648665 (99m:13040)
13. G.-M. Greuel, G. Pfister, and H. Schönemann.
SINGULAR 3.0.
A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2005.
14. D. Lazard, Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations, Computer algebra (London, 1983) Lecture Notes in Comput. Sci., vol. 162, Springer, Berlin, 1983, pp. 146–156. MR 774807 (86m:13002)
15. Daniel Lazard, Résolution des systèmes d’équations algébriques, Theoret. Comput. Sci. 15 (1981), no. 1, 77–110 (French, with English summary). MR 619687 (82i:12001), http://dx.doi.org/10.1016/0304-3975(81)90064-5

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