### 7.7.2 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:

> 7.7.2.0. bfct |  | compute the BS polynomial of f with linear reductions
> 7.7.2.0. bfctSyz |  | compute the BS polynomial of f with syzygy-solver
> 7.7.2.0. bfctAnn |  | compute the BS polynomial of f via Ann f^s + f
> 7.7.2.0. bfctOneGB |  | compute the BS polynomial of f by just one GB computation
> 7.7.2.0. bfctIdeal |  | compute the b-function of ideal w.r.t. weight
> 7.7.2.0. pIntersect |  | intersection of ideal with subalgebra K[f] for a polynomial f
> 7.7.2.0. pIntersectSyz |  | intersection of ideal with subalgebra K[f] with syz-solver
> 7.7.2.0. linReduce |  | reduce a polynomial by linear reductions w.r.t. ideal
> 7.7.2.0. linReduceIdeal |  | interreduce generators of ideal by linear reductions
> 7.7.2.0. linSyzSolve |  | compute a linear dependency of elements of ideal I
> 7.7.2.0. allPositive |  | checks whether all entries of an intvec are positive
> 7.7.2.0. scalarProd |  | computes the standard scalar product of two intvecs
> 7.7.2.0. vec2poly |  | constructs an univariate polynomial with given coefficients

See also: dmod_lib; dmodapp_lib; dmodvar_lib; gmssing_lib.

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