Localization at hyperplane arrangements: combinatorics and \({\mathcal D}\)-modules. (English) Zbl 1135.32010
Some results of the first author from a joint paper with R. Garcia López and S. A. Zarzuela [Adv. Math. 174, No. 1, 35–56 (2003; Zbl 1050.13009)] are combined with a method developed by the second and the third author [J. Symb. Comput. 32, No. 6, 677–685 (2001; Zbl 1015.16029); ibid. 41, No. 3–4, 317–335 (2006; Zbl 1126.16015)] for to describe a new interesting algorithm deciding whether the annihilating ideal of the meromorphic function \(1/f\), where \(f=0\) defines an arrangement of hyperplanes generated by linear differential operators of order 1. The mentioned algorithm makes comparison of two characteristic cycles. The notion of a characteristic cycle of a holonomic \(\Lambda\)-module \(M\) is presented in a very sophisticated way with complicated notations based on other auxiliaries notions like characteristic variety \(\text{CH}(M)\) and characteristic ideal \({\mathcal J}(M)\) (better to see the paper under review). Comments related to the known example of Orlik-Teraos hyperplane arrangements are given [see P. Orlik and H. Terao, Math. Ann. 301, No. 2, 211–235 (1995; Zbl 0813.32033)]. Direct computations of the annihilating ideal of \(1/f\), \(f=0\), are discussed, too.
Reviewer: Stanco Dimiev (Sofia)
MSC:
| 32C38 | Sheaves of differential operators and their modules, \(D\)-modules |
| 35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |
Keywords:
characteristic cycle; localization; hyperplane arrangements; logarithmic \(\mathcal D\)-modules; Gröbner bases; ideals of meromorphic function; linear differential operators of order 1; holonomic \(\Lambda\)-modulesReferences:
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