
7.5.2 bfun_lib
 Library:
 bfun.lib
 Purpose:
 Algorithms for bfunctions and BernsteinSato polynomial
 Authors:
 Daniel Andres, daniel.andres@math.rwthaachen.de
Viktor Levandovskyy, levandov@math.rwthaachen.de
 Overview:
 Given a polynomial ring R = K[x_1,...,x_n] and a polynomial F in R,
one is interested in the global bfunction (also known as BernsteinSato
polynomial) b(s) in K[s], defined to be the nonzero 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 nth 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:
 BernsteinSato 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 bfunction 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
Kvector 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 bfunction of I w.r.t. w.
Unlike BernsteinSato polynomial, general bfunction with respect to
arbitrary weights need not have rational roots at all. However, bfunction
of a holonomic ideal is known to be nonzero as well.
 References:
 [SST] Saito, Sturmfels, Takayama: Groebner Deformations of
Hypergeometric Differential Equations (2000),
Noro: An Efficient Modular Algorithm for Computing the Global bfunction,
(2002).
Procedures:
See also:
dmod_lib;
dmodapp_lib;
dmodvar_lib;
gmssing_lib.
