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.
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