zbMATH — the first resource for mathematics

Computing $$\mu^{*}$$-sequences of hypersurface isolated singularities via parametric local cohomology systems. (English) Zbl 1393.13035
Let $$X \subset \mathbb{C}^n$$ be an open neighborhood of the origin $$O$$ and $$f$$ a holomorphic function defined on $$X$$ with isolated singularity at $$O$$. The Milnor number $$\mu(f)$$ is the length of $$\mathcal O_{X,O}/J(f)$$ where $$J(f)$$ is the Jacobi ideal. For $$i = 0$$, …, $$n$$, let $\mu^{(i)}(f) = \min_L \mu(f|_L)$ where $$L$$ runs over all the $$i$$-dimensional subspaces of $$\mathbb{C}^n$$. The $$\mu^*$$-sequence is defined to be $\mu^*(f) = (\mu^{(n)}(f), \dots, \mu^{(0)}(f)).$ In the present paper, authors give a new algorithm to compute $$\mu^*(f)$$. It is based on author’s preceding paper [in: Proceedings of the 39th international symposium on symbolic and algebraic computation, ISSAC 2014, Kobe, Japan, July 23–25, 2014. New York, NY: Association for Computing Machinery (ACM). 351–358 (2014; Zbl 1325.68297)]. Furthermore this paper contains a table of $$\mu^*$$-sequence of famous singularties.

MSC:
 13D45 Local cohomology and commutative rings 32C37 Duality theorems for analytic spaces 13J05 Power series rings 32A27 Residues for several complex variables
Risa/Asir
Full Text:
References:
 [1] Biviá-Ausina, C, Generic linear sections of complex hypersurfaces and monomial ideals, Topology Appl., 159, 414-419, (2012) · Zbl 1238.32019 [2] Briançon, J; Speder, JP, La trivialité topologique n’implique pas LES conditions de Whitney, C. R. Acad. Sci. Paris, 280, 365-367, (1975) · Zbl 0331.32010 [3] Briançon, J; Henry, JP; Speder, JP, LES conditions de Whitney en un point sont analytiques, C. R. Acad. Sci. Paris, 282, 279-282, (1976) · Zbl 0331.32011 [4] Briançon, J; Speder, JP, LES conditions de Whitney impliques μ∗ constant, Ann. Inst. Fourier Grenoble, 26, 153-163, (1976) · Zbl 0331.32012 [5] Briançon, J; Henry, JPG, Equisingularité générique des familles de surfaces a singularité isolée, Bull. Soc. Math. France, 108, 259-281, (1980) · Zbl 0482.14004 [6] Gaffney, T, Polar multiplicities and equisingularity of map germs, Topology, 32, 185-223, (1993) · Zbl 0790.57020 [7] Grothendieck, A.: Théorèmes de dualité pour les faisceaux algébriques cohérents. Séminaire Bourbaki 149, 25 pp (1957) [8] Grothendieck, A.: Local Cohomology, Notes by R. Hartshorne. Lecture Notes in Math., vol. 41. Springer (1967) · Zbl 0185.49202 [9] Kouchnirenko, AG, Polyèdres de Newton et nombres de Milnor, Invent. Math., 32, 1-31, (1976) · Zbl 0328.32007 [10] Mima, S, On the Milnor number of a generic hyperplane section, J. Math. Soc. Jpn., 41-4, 709-724, (1989) · Zbl 0686.32004 [11] Nabeshima, K., Tajima, S.: On efficient algorithm for computing parametric local cohomology classes associated with semi-quasihomogeneous singularities and standard bases. In: Proc. International Symposium on Symbolic and Algebraic Computation (ISSAC 2014), pp. 351-358. ACM (2014) · Zbl 1325.68297 [12] Nabeshima, K., Tajima, S.: Algebraic local cohomology with parameters and parametric standard bases for zero-dimensional ideals. To appear in Journal of Symbolic Computation (2016) · Zbl 1430.13026 [13] Navarro Azunar, V, Conditions de Whitney et sections planes, Invent. Math., 61, 199-225, (1980) · Zbl 0449.32013 [14] Noro, M., Takeshima, T.: Risa/Asir - A computer algebra system. In: Proc. International Symposium on Symbolic and Algebraic Computation(ISSAC1992), pp. 387-396. ACM. http://www.math.kobe-u.ac.jp/Asir/asir.html (1992) · Zbl 0964.68597 [15] O’Shea, D., Teleman, C.: Limiting tangent spaces and a criterion for $$μ$$-constancy. Travaux en Cours 55 Hermann, 79-85 (1997) · Zbl 1238.32019 [16] Tajima, S; Nakamura, Y; Nabeshima, K, Standard bases and algebraic local cohomology for zero dimensional ideals, Adv. Stud. Pure Math., 56, 341-361, (2009) · Zbl 1194.13020 [17] Teissier, B, Cycles évanescents et conditions de Whitney, C. R. Acad. Sc. Paris, 276, 1051-1054, (1973) · Zbl 0263.32011 [18] Teissier, B.: Cycles evanescents, sections planes et conditions de Whitney. Astérisques 7-8, Soc. Math. France, 285-362 (1973) · Zbl 0295.14003 [19] Teissier, B, Variétés polaires I, Invent. Math., 40, 267-292, (1977) · Zbl 0446.32002 [20] Teissier, B, Variétés polaires II, Lect. Notes Math., 961, 314-491, (1982) · Zbl 0585.14008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.