Computation of Bivariate Characteristic Polynomials of Finitely Generated Modules over Weyl Algebras

TL;DR

This paper extends Groebner basis techniques to compute bivariate dimension polynomials for finitely generated modules over Weyl algebras, revealing new invariants and applications in D-module isomorphism and differential algebraic groups.

Contribution

It introduces a method to compute bivariate dimension polynomials for D-modules, highlighting invariants not captured by Bernstein polynomials.

Findings

Developed algorithms for computing bivariate dimension polynomials.

Showed that these polynomials contain additional invariants.

Applied results to D-module isomorphism and classification of algebraic groups.

Abstract

In this paper we generalize the classical Groebner basis technique to prove the existence and present a method of computation of a dimension polynomial in two variables associated with a finitely generated D-module, that is, a finitely generated module over a Weyl algebra. We also present corresponding algorithms and examples of computation of such polynomials and show that a bivariate dimension polynomial can contain some invariants that are not carried by the Bernstein dimension polynomial. The obtained results are applied to the isomorphism problem for $D$-modules; they have also potential applications to classification problems of differential algebraic groups.