  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. This paper develops a novel structure theory for primary
ideals in a polynomial ring. We characterize primary ideals in terms of PDE,
punctual Hilbert schemes, relative Weyl algebras, and the join construction.
Solving the PDE described by a primary ideal amounts to computing Noetherian
operators in the sense of Ehrenpreis and Palamodov. We develop new algorithms
for this task, and we present efficient implementations.
