Singular

Dear Singular and Singular:Plural users,

I am glad to announce, that the library '[b]dmod.lib[/b]', containing the algorithms for algebraic D-modules, has been released and will be distributed with Singular:Plural starting from the version 3-0-2.

The library has been created by Viktor Levandovskyy (RISC, Linz, Austria) and Jorge Martin Morales (Zaragoza, Spain). With this release we provide the implementation of several fundamental algorithms in D-module theory as well as some helpful tools.

[b]Functionality.[/b] Namely, there are two flavors of the Ann F^s algorithm ([b]annfsOT [/b]and [b]annfsBM[/b]; see details below). In particular, as a byproduct of annfs* procedures, the Bernstein polynomial is computed. Also, we provide the procedures for setting nontrivial examples ([b]reiffen [/b]for Reiffen curves and [b]arrange [/b]for hyperplane arrangements). With the help of Gelfand-Kirillov dimension from [b]gkdim.lib[/b], we provide the procedure [b]isHolonomic[/b], checking whether a given module is holonomic.

Since both annfs algorithms are using complicated elimination, a powerful Groebner basis algorithm is needed. We provide a flexible possibility to choose, whether the whole computations will use either [b]std [/b]or [b]slimgb [/b]- please consult with the documentation.

[b]Future development.[/b] We see this release as the first stone in the fundament of a powerful library for D-modules theory, comparable in the functionality with the implementations in Macaulay2 and kan/sm1.
Moreover, we plan to follow the ideas of Bahloul and Ucha and implement the algorithms for Berstein-Sato ideals of products of polynomials.
We are open to cooperation with anybody who wish to add some functionality to the library. Please contact me on the recent state of the development.

[b]Some theory.[/b] Given a polynomial ring R = K[x_1,...,x_n] and a polynomial F in R, one is interested in the ring R[1/F^s] for a natural number s. In fact, the ring R[1/F^s] has a structure of a D(R)-module, where D(R) is a Weyl algebra K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1, d_k x_i = x_i d_k, i!=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^s] is isomorphic to D(R)/I as a D(R)-module. We provide the implementation of two algorithms:
1) the classical Ann F^s algorithm from Oaku and Takayama (J. Pure Applied Math., 1999),
2) the newer Ann F^s algorithm by Briancon and Maisonobe (Remarques sur lÂ’ideal de Bernstein associe a des polynomes, preprint, 2002).

[b]Documentation.[/b] All the procedures in the library are well-documented and equipped with nontrivial examples. Setting the [i]printlevel [/i]to 2, all the technical messages will be printed. In particular, each step of the algorithm is traced and the intermediate results are printed out. Alternatively, setting the [i]printlevel [/i]to 1 you will get only the progress messages, shortly commenting the computations.

[b]Citation.[/b] If you're using the library, please cite both Singular:Plural and the library (separately, if it is possible). Here is the corresponding BiBTeX entry for the library:

@Article{dmodlib,
author = {{Levandovskyy V. and Morales, J.}},
title = {{A textsc{Singular} 3.0 library for computations with algebraic D-modules texttt{dmod.lib}}},
year = {{2006}}
}

Have fun with the library,

Viktor Levandovskyy