### 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:

> 7.5.6.0. annfsLogIdeal |  | compute the logarithmic annihilator Ann^(1) F^s in D
> 7.5.6.0. annihilatorMultiFs |  | compute the annihilator Ann F^s in D
> 7.5.6.0. BSidealFromAnn |  | compute several kinds of Bernstein-Sato ideals, given Ann F^s
> 7.5.6.0. BernsteinSatoIdeal |  | compute several kinds of Bernstein-Sato ideals, given only F
> 7.5.6.0. BFBoundsBudur |  | compute upper and lower bounds of several kinds of Bernstein-Sato ideals with the method of (Bud12)
> 7.5.6.0. annfalphaI |  | compute Ann F^alpha in D, where alpha is an ideal from the ground field
> 7.5.6.0. extractS |  | give I as ideal in the commutative polynomial ring in the first r variables

See also: bfun_lib; dmod_lib; dmodapp_lib; dmodloc_lib; gmssing_lib.

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