|
7.5.2 dmod_lib
- Library:
- dmod.lib
- Purpose:
- Algorithms for algebraic D-modules
- Authors:
- Viktor Levandovskyy, levandov@risc.uni-linz.ac.at
Jorge Martin Morales, jorge@unizar.es
- Theory:
- Given a polynomial ring R = K[x_1,...,x_n] and a polynomial F in R,
one is interested in the R[1/F]-module of rank one, generated by F^s
for a natural number s.
In fact, the module R[1/F]*F^s has a structure of a D(R)[s]-module, where D(R)
is an n-th Weyl algebra K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1> and
D(R)[s] = D(R) tensored with K[s] over K.
Constructively, one needs to find a left ideal I = I(F^s) in D(R), such
that K[x_1,...,x_n,1/F]*F^s is isomorphic to D(R)/I as a D(R)-module.
We often write just D for D(R) and D[s] for D(R)[s].
One is interested in the following data:
- Ann F^s = I = I(F^s) in D(R)[s], denoted by LD in the output
- global Bernstein polynomial in K[s], denoted by bs, its minimal integer root s0 and
the list of all roots of bs, which are rational, with their multiplicities is denoted by BS
- Ann F^s0 = I(F^s0) in D(R), denoted by LD0 in the output (LD0 is a holonomic ideal in D(R))
- Ann^(1) F^s in D(R)[s], denoted by LD1 (logarithmic derivations)
- an operator in D(R)[s], denoted by PS, such that PS*F^(s+1) = bs*F^s holds in K[x_1,...,x_n,1/F^s].
We provide the following implementations:
OT) the classical Ann F^s algorithm from Oaku and Takayama (J. Pure
Applied Math., 1999),
LOT) Levandovskyy's modification of the Oaku-Takayama algorithm (unpublished)
BM) the Ann F^s algorithm by Briancon and Maisonobe (Remarques sur
l'ideal de Bernstein associe a des polynomes, preprint, 2002)
- Guide:
- Ann F^s = I = I(F^s) = LD in D(R)[s] can be computed by SannfsBM, SannfsOT, SannfsLOT
- Ann^(1) F^s in D(R)[s] can be computed by Sannfslog
- global Bernstein polynomial bs resp. BS in K[s] can be computed by bernsteinBM
- Ann F^s0 = I(F^s0) = LD0 in D(R) can be computed by annfs0, annfsBM, annfsOT, annfsLOT
- all the relevant data (LD, LD0, bs, PS) are computed by operatorBM
Main procedures:
7.5.2.0. annfs | | compute Ann F^s0 in D and Bernstein poly for a poly F |
7.5.2.0. annfspecial | | compute Ann F^n from Ann F^s for a poly F and a number n |
7.5.2.0. Sannfs | | compute Ann F^s in D[s] for a poly F |
7.5.2.0. Sannfslog | | compute Ann^(1) F^s in D[s] for a poly F |
7.5.2.0. bernsteinBM | | compute global Bernstein poly for a poly F (algorithm of Briancon-Maisonobe) |
7.5.2.0. operatorBM | | compute Ann F^s, Ann F^s0, BS and PS for a poly F (algorithm of Briancon-Maisonobe) |
7.5.2.0. annfsParamBM | | compute the generic Ann F^s (algorithm by Briancon and Maisonobe) and exceptional parametric constellations for a poly F with parametric coefficients |
7.5.2.0. annfsBMI | | compute Ann F^s and Bernstein ideal for a poly F=f1*..*fP (multivariate algorithm of Briancon-Maisonobe) |
7.5.2.0. checkRoot | | check if a given rational is a root of the global Bernstein polynomial of F and compute its multiplicity SECONDARY PROCEDURES FOR D-MODULES: |
7.5.2.0. annfsBM | | compute Ann F^s0 in D and Bernstein poly for a poly F (algorithm of Briancon-Maisonobe) |
7.5.2.0. annfsLOT | | compute Ann F^s0 in D and Bernstein poly for a poly F (Levandovskyy modification of the Oaku-Takayama algorithm) |
7.5.2.0. annfsOT | | compute Ann F^s0 in D and Bernstein poly for a poly F (algorithm of Oaku-Takayama) |
7.5.2.0. SannfsBM | | compute Ann F^s in D[s] for a poly F (algorithm of Briancon-Maisonobe) |
7.5.2.0. SannfsLOT | | compute Ann F^s in D[s] for a poly F (Levandovskyy modification of the Oaku-Takayama algorithm) |
7.5.2.0. SannfsOT | | compute Ann F^s in D[s] for a poly F (algorithm of Oaku-Takayama) |
7.5.2.0. annfs0 | | compute Ann F^s0 in D and Bernstein poly from the known Ann F^s in D[s] |
7.5.2.0. checkRoot1 | | check whether a rational is a root of the global Bernstein polynomial of F from the known Ann F^s in D[s] |
7.5.2.0. checkRoot2 | | check whether a rational is a root of the global Bernstein polynomial of F and compute its multiplicity from the known Ann F^s in D[s] |
7.5.2.0. checkFactor | | check whether a polynomial qs in K[s] is a factor of the global Bernstein polynomial of F from the known Ann F^s in D[s] |
Auxiliary procedures:
gmssing_lib
|