Castro-Jiménez, F. J.; Ucha-Enríquez, J. M. Testing the logarithmic comparison theorem for free divisors. (English) Zbl 1071.14024 Exp. Math. 13, No. 4, 441-449 (2004). Summary: We propose in this work a computational criterion to test if a free divisor \(D\subset\mathbb{C}^n\) verifies the logarithmic comparison theorem (LCT); that is, whether the complex of logarithmic differential forms computes the cohomology of the complement of \(D\) in \(\mathbb{C}^n\). For Spencer free divisors \(D\equiv(f=0)\), we solve a conjecture about the generators of the annihilating ideal of \(1/f\) and make a conjecture on the nature of Euler homogeneous free divisors which verify LCT. In addition, we provide examples of free divisors defined by weighted homogeneous polynomials that are not locally quasi-homogeneous. Cited in 7 Documents MSC: 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14F40 de Rham cohomology and algebraic geometry 32C38 Sheaves of differential operators and their modules, \(D\)-modules 68W30 Symbolic computation and algebraic computation 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Keywords:de Rham cohomology; Gröbner bases; \({\mathcal D}\)-modules; Spencer divisors Software:Risa/Asir; D-modules; Macaulay2 PDFBibTeX XMLCite \textit{F. J. Castro-Jiménez} and \textit{J. M. Ucha-Enríquez}, Exp. Math. 13, No. 4, 441--449 (2004; Zbl 1071.14024) Full Text: DOI Euclid EuDML Link References: [1] Bernstein J., Funkcional. Anal, i Prilozen 6 (4) pp 26– (1972) [2] Calderón-Moreno F. J., Ann. Sci. E.N.S. 32 (4) pp 701– (1999) [3] DOI: 10.1007/s00014-002-8330-6 · Zbl 1010.32016 [4] DOI: 10.1023/A:1020228824102 · Zbl 1017.32023 [5] Calderón-Moreno F. J., ”Dualité et comparaison sur les complexes de de Rham logarithmiques par rapport aux di-viseurs libres.” (2004) [6] DOI: 10.1090/S0002-9947-96-01690-X · Zbl 0862.32021 [7] DOI: 10.1006/jsco.2001.0489 · Zbl 1015.16029 [8] Castro Jiméne F. J., Proc. Steklov Inst, of Math. 238 pp 88– (2002) [9] Grayson D., ”Macaulay2: A Software System for Research in Algebraic Geometry.” (1999) [10] Grothendieck A., Publ. Math, de l’I.H.E.S. 29 pp 95– (1966) [11] Leykin A., ”D-Module Package for Macaulay 2.” (2001) [12] Mebkhout Z., Le formalisme des six opérations de Grothendieck pour les Dx-modules cohérents (1989) [13] Noro M., Mathematical Software (Beijing, 2002) pp 147– (2002) [14] Noro M., ”A Computer Algebra System Risa/Asir.” (2000) · Zbl 1027.68152 [15] DOI: 10.1016/S0022-4049(97)00024-8 · Zbl 0918.32006 [16] Sato M., Hypergeometric and Pseudo-Differential Equations pp 265– (1973) [17] Saito K., J. Fac Sci. Univ. Tokyo 27 pp 256– (1980) [18] Saito M., Gröbner Deformations of Hypergeometric Differential Equations (2000) [19] Torrelli T., PhD. diss., in: ”Equations fonctionnelles pour une fonction sur un espace singulier.” (1998) [20] Torrelli T., Bull. Soc. Math. France. (2004) [21] Ucha-Enríquez J. M., PhD. diss., in: ”Métodos constructivos enálgebras de operadores diferenciales.” (1999) 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.