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Introduction to algorithms for \(D\)-modules with quiver \(D\)-modules. (English) Zbl 1453.14057
Iohara, Kenji (ed.) et al., Two algebraic byways from differential equations: Gröbner bases and quivers. Cham: Springer. Algorithms Comput. Math. 28, 95-114 (2020).
Summary: The goal of this expository chapter is to illustrate how to use algorithmic methods for \(D\)-modules to make mathematical experiments for \(D\)-modules and cohomology groups with examples of quiver \(D\)-modules. The first section is based on a lecture by the second author given in the Kobe-Lyon summer school 2015 On Quivers: s Computational Aspects and Geometric Applications. The second author could attend several interesting lectures of the school and the Sects. 2 and 3 are written by an inspiration from these lectures and the interesting paper by S. Khoroshkin and A. Varchenko [IMRP, Int. Math. Res. Pap. 2006, No. 20, Article ID 69590 116 p. (2006; Zbl 1116.32005)].
For the entire collection see [Zbl 1444.14002].
MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C38 Sheaves of differential operators and their modules, \(D\)-modules
13N10 Commutative rings of differential operators and their modules
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14F40 de Rham cohomology and algebraic geometry
Software:
Risa/Asir
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References:
[1] R. Bahloul, Gröbner bases in \(D\)-modules: applications to Bernstein-Sato polynomials, in Two Algebraic Byways from Differential Equations (Gröbner Bases and Quivers. Algorithms and Computation in Mathematics (Springer, Berlin, 2019).
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[4] K. Iohara, Introduction to representations of quivers, in Two Algebraic Byways from Differential equations: (Gröbner Bases and Quivers. Algorithms and Computation in Mathematics (Springer, Berlin, 2019).
[5] S. Khoroshkin, A. Varchenko, Quiver \(\mathscr{D} \)-modules and homology of local systems over an arrangement of hyperplanes. IMRP Int. Math. Res. Pap. Art. ID 69590, 116 (2006). arxiv: math/0510451v2. · Zbl 1116.32005
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[9] T. Oaku, N. Takayama, An algorithm for de Rham cohomology groups of the complement of an affine variety via \(D\)-module computation. J. Pure Appl. Algebra 139(1-3), 201-233 (1999). Effective methods in algebraic geometry (Saint-Malo, 1998). · Zbl 0960.14008
[10] T. Oaku, N. Takayama, Algorithms for \(D\)-modules-restriction, tensor product, localization, and local cohomology groups. J. Pure Appl. Algebra 156(2-3), 267-308 (2001). · Zbl 0983.13008
[11] Risa/Asir. A Computer Algebra System.
[12] M.
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