Anton Leykin

D-MODULES FOR MACAULAY 2

The Macaulay 2 D-modules package is an implementation of the Weyl algebra and algorithms related to it in the computer algebra system Macaulay 2. Over the last decade there were many advances made in the computational theory of D-modules. Several newest algorithmic methods used in the analysis of hypergeometric differential equations are described in the recent book (). Also we worked with a paper by Oaku and Takayama () providing, in particular, a detailed description of the restriction algorithm. The algorithms for computing localization, D-homomorphisms, and global b-functions of polynomials with parameters come from (), (), and () respectively. As to the applications that we describe here, we compute polynomial and rational solutions according to (), local cohomology via Cech complex comes from a paper of Walther (). Macaulay 2, a noncommercial computer algebra system crafted by Grayson and Stillman, became one of the favorite tools for specialists in algebraic analysis. You are welcome to join the crowd by downloading the current distribution from the web (see ()). D-modules package comes as a part of it, for the most recent updates and online documentation see (). If you would like to learn more about Macaulay 2, read a recently published book (), which contains a set of very interesting examples of computations in algebraic geometry including a section by Walther featuring D-modules package. At the end of the introduction, let us mention that there are several other systems that are capable of handling D-modules. First on the list is Takayama’s system Kan (), which is a specialized system for Dcomputations. There is an implementation of Weyl algebra in Maple by Chyzak (). It is also implemented in Singular (), although not included in the main package.