Primary ideals and their differential equations

An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. The primary ideals arising in such a decomposition can be characterized in terms of certain differential equations. In this talk we will understand how this characterization works. We will give an explicit algorithm for computing these differential operators that describe a primary ideal, namely Noetherian operators. For some special cases, we will give an alternative representation of the primary ideal by differential equations playing with the join construction. This is a joint work with Yairon Cid-Ruiz and Bernd Sturmfels.