Department of Mathematics Server University of Manitoba

Abstract: The first Weyl algebra over a field K acts naturally on the ring of polynomials K[x] by multiplication and formal differentiation. As such, K[x] is a uniform module, and so its injective envelope, the module of formal solutions to linear differential equations over polynomials, is an indecomposable injective module. What does E(K[x]) look like? Is there an ``explicit description'' (in the admittedy rather vague sense of some of my papers)? Can we find a description in terms of ``inverse polynomials''? How much does the structure of E(K[x]) depend on the structure of the field K? Can we represent E(C[x]) as holomorphic functions on (uncomplicated) subsets of C?

The answers to all of these questions might very well be ``no'', that is, E(K[x]) might be provably complicated. I will present some very preliminary computations that indicate some of the internal structural complexity of this injective module.