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7.5.6 dmodideal_lib
- Library:
- dmodideal.lib
- Purpose:
- Algorithms for Bernstein-Sato ideals of morphisms
- Authors:
- Robert Loew, robert.loew at rwth-aachen.de
Viktor Levandovskyy, levandov at math.rwth-aachen.de
Jorge Martin Morales, jorge at unizar.es
- Overview:
- Let K be a field of characteristic 0. Given a polynomial ring
R = K[x_1,...,x_n] and a map, given by polynomials F_1,...,F_r from R,
one is interested in the R[1/(F_1*...*F_r)]-module of rank one, generated by
the symbol F^s=F_1^(s_1)*...*F_r^(s_r) for symbolic discrete variables s_1,...,s_r.
This module R[1/(F_1*...*F_r)]*F^s has a structure of a D(R)[s_1,...,s_r]-module, where D(R)
is an n-th Weyl algebra K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1> and
D(R)[s] := D(R) tensored with K[s]:=K[s_1,...,s_r] over K.
We often write just D for D(R) and D[s] for D(R)[s].
One is interested in the computation of the following data:
- Ann_{D[s]} F^s, the annihilator of F^s in D[s]; see annihilatorMultiFs
- Ann^{1}_{D[s]} F^s, the logarithmic annihilator of F^s in D[s]; see annfsLogIdeal
- several kinds of global Bernstein-Sato ideals in K[s],
cf. (CU) and (Bud12); see BernsteinSatoIdeal and BSidealFromAnn
- Ann_{D} F^alpha for alpha from K^r, the annihilator of F^alpha in D; see annfalphaI
- sub- and over-ideals, bounding the Bernstein-Sato ideal; see BFBoundsBudur
- References:
- (BM) the Ann F^s algorithm by Briancon and Maisonobe (Remarques sur
l'ideal de Bernstein associe a des polynomes, preprint, 2002)
(LM08) V. Levandovskyy and J. Martin-Morales, ISSAC 2008
(CU) Castro and Ucha, On the computation of Bernstein-Sato ideals, JSC 2005
(SST) Saito, Sturmfels, Takayama 'Groebner Deformations of Hypergeometric
Differential Equations', Springer, 2000
(Bud12) N. Budur, Bernstein-Sato ideals and local systems, Annales de
l'Institut Fourier, Volume 65 (2015) no. 2
(OT99) T. Oaku and N. Takayama, An algorithm for de Rham cohomology groups of
the complement of an affine variety via D-module computation,
Journal of Pure and Applied Algebra, 1999
Procedures:
See also:
bfun_lib;
dmod_lib;
dmodapp_lib;
dmodloc_lib;
gmssing_lib.
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