D-modules and holonomic functions

Abstract

In algebraic geometry, one studies the solutions to polynomial equations or, equivalently, to linear partial differential equations with constant coefficients. These lecture notes address the more general case when the coefficients of the partial differential equations are polynomials. The letter $D$ stands for the Weyl algebra, and a $D$-module is a left module over $D$. We focus on left ideals or $D$-ideals. We represent holonomic functions in several variables by the linear differential equations they satisfy. This encoding by a $D$-ideal is useful for many problems, e.g., in geometry, physics and statistics. We explain how to work with holonomic functions. Applications include volume computations and likelihood inference.