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7.5.2 bfun_lib

Library:
bfun.lib
Purpose:
Algorithms for b-functions and Bernstein-Sato polynomial
Authors:
Daniel Andres, daniel.andres@math.rwth-aachen.de
Viktor Levandovskyy, levandov@math.rwth-aachen.de

Overview:
Given a polynomial ring R = K[x_1,...,x_n] and a polynomial F in R, one is interested in the global b-function (also known as Bernstein-Sato polynomial) b(s) in K[s], defined to be the non-zero monic polynomial of minimal degree, satisfying a functional identity L * F^{s+1} = b(s) F^s, for some operator L in D[s] (* stands for the action of differential operator)
By D one denotes the n-th Weyl algebra
K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1>.
One is interested in the following data:
- Bernstein-Sato polynomial b(s) in K[s],
- the list of its roots, which are known to be rational
- the multiplicities of the roots.

There is a constructive definition of a b-function of a holonomic ideal I in D (that is, an ideal I in a Weyl algebra D, such that D/I is holonomic module) with respect to the given weight vector w: For a polynomial p in D, its initial form w.r.t. (-w,w) is defined as the sum of all terms of p which have maximal weighted total degree where the weight of x_i is -w_i and the weight of d_i is w_i. Let J be the initial ideal of I w.r.t. (-w,w), i.e. the K-vector space generated by all initial forms w.r.t (-w,w) of elements of I. Put s = w_1 x_1 d_1 + ... + w_n x_n d_n. Then the monic generator b_w(s) of the intersection of J with the PID K[s] is called the b-function of I w.r.t. w. Unlike Bernstein-Sato polynomial, general b-function with respect to arbitrary weights need not have rational roots at all. However, b-function of a holonomic ideal is known to be non-zero as well.

References:
[SST] Saito, Sturmfels, Takayama: Groebner Deformations of Hypergeometric Differential Equations (2000),
Noro: An Efficient Modular Algorithm for Computing the Global b-function, (2002).

Procedures:

7.5.2.0. bfct  compute the BS polynomial of f with linear reductions
7.5.2.0. bfctSyz  compute the BS polynomial of f with syzygy-solver
7.5.2.0. bfctAnn  compute the BS polynomial of f via Ann f^s + f
7.5.2.0. bfctOneGB  compute the BS polynomial of f by just one GB computation
7.5.2.0. bfctIdeal  compute the b-function of ideal w.r.t. weight
7.5.2.0. pIntersect  intersection of ideal with subalgebra K[f] for a polynomial f
7.5.2.0. pIntersectSyz  intersection of ideal with subalgebra K[f] with syz-solver
7.5.2.0. linReduce  reduce a polynomial by linear reductions w.r.t. ideal
7.5.2.0. linReduceIdeal  interreduce generators of ideal by linear reductions
7.5.2.0. linSyzSolve  compute a linear dependency of elements of ideal I
7.5.2.0. allPositive  checks whether all entries of an intvec are positive
7.5.2.0. scalarProd  computes the standard scalar product of two intvecs
7.5.2.0. vec2poly  constructs an univariate polynomial with given coefficients
See also: dmod_lib; dmodapp_lib; dmodvar_lib; gmssing_lib.