19 - Gröbner Bases in Rings of Differential Operators

Summary

Introduction

Let R be the ring of all complex rational functions without poles in a given real interval. The work of U. Oberst and S. Fröhler ([7],[8],[14]) on systems of differential equations with time-varying coefficients raised several questions for modules over the ring R[D] of linear differential operators with coefficients in R.

There are a number of results ([2],[5],[6],[9],[11],[12],[13],[17],…) on Gröbner bases in rings of differential operators, but the coefficient rings are fields (of rational functions), rings of power series, or rings of polynomials over a field. In the latter case every differential operator is a K-linear combination of “terms” xⁱ D_j, (i,j) ∈ Nⁿ × Nⁿ. Thus Gröbner bases are defined with respect to a term order on Nⁿ × Nⁿ, the coefficients are elements of a field and commute with the terms. This approach cannot be used for other coefficient rings (like R, for example).

The results of B. Buchberger ([3],[4]) on Gröbner bases in polynomial rings have been generalized by several authors (see for example [9]) to polynomial rings with coefficients in commutative rings. In analogy to this extension we present a basic theory of Gröbner bases for differential operators with coefficients in a commutative ring.