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7.7.2 bfun_lib
- Library:
- bfun.lib
- Purpose:
- Algorithms for b-functions and Bernstein-Sato polynomial
- Authors:
- Daniel Andres, [email protected]
Viktor Levandovskyy, [email protected]
- 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:
See also:
dmod_lib;
dmodapp_lib;
dmodvar_lib;
gmssing_lib.
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