          ### 7.5.7 dmodvar_lib

Library:
dmodvar.lib
Purpose:
Algebraic D-modules for varieties

Authors:
Daniel Andres, daniel.andres@math.rwth-aachen.de
Viktor Levandovskyy, levandov@math.rwth-aachen.de
Jorge Martin-Morales, jorge@unizar.es

Support: DFG Graduiertenkolleg 1632 'Experimentelle und konstruktive Algebra'

Overview:
Let K be a field of characteristic 0. Given a polynomial ring R = K[x_1,...,x_n] and polynomials f_1,...,f_r in R, define F = f_1*...*f_r and F^s = f_1^s_1*...*f_r^s_r for symbolic discrete (that is shiftable) variables s_1,..., s_r. The module R[1/F]*F^s has the structure of a D<S>-module, where D<S> = D(R) tensored with S over K, where
- D(R) is the n-th Weyl algebra K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j + 1>
- S is the universal enveloping algebra of gl_r, generated by s_i = s_{ii}.
One is interested in the following data:
- the left ideal Ann F^s in D<S>, usually denoted by LD in the output
- global Bernstein polynomial in one variable s = s_1+...+s_r, denoted by bs,
- its minimal integer root s0, the list of all roots of bs, which are known to be negative rational numbers, with their multiplicities, which is denoted by BS
- an r-tuple of operators in D<S>, denoted by PS, such that the functional equality sum(k=1 to k=r) P_k*f_k*F^s = bs*F^s holds in R[1/F]*F^s.

References:
(BMS06) Budur, Mustata, Saito: Bernstein-Sato polynomials of arbitrary varieties (2006).
(ALM09) Andres, Levandovskyy, Martin-Morales: Principal Intersection and Bernstein-Sato Polynomial of an Affine Variety (2009).

Procedures:

 7.5.7.0. bfctVarIn computes the roots of the Bernstein-Sato polynomial b(s) of the variety V(F) using initial ideal approach 7.5.7.0. bfctVarAnn computes the roots of the Bernstein-Sato polynomial b(s) of the variety V(F) using Sannfs approach 7.5.7.0. SannfsVar computes the annihilator of F^s in the ring D 7.5.7.0. makeMalgrange creates the Malgrange ideal, associated with F = F,..,F[P]          