Linear PDE with Constant Coefficients

In an undergraduate differential equations course we learn to solve a homogeneous linear ordinary differential equation with constant coefficients by finding roots of the characteristic polynomial. Thus the problem of solving an ODE is reduced to factoring a univariate polynomial. A generalization of this was found in the 1960s for systems of linear PDE. The Fundamental Theorem by Ehrenpreis and Palamodov asserts that solutions to a PDE system can be represented by a finite sum of integrals over some algebraic variety. This representation can be used describe a primary decomposition of an ideal or module. This talk presents the main historical results, along with recent algorithms.