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.

中文翻译：

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基本理想及其微分方程

多项式环中的理想值对具有常数系数的线性偏微分方程组进行编码。初级分解组织了PDE的解决方案。本文针对多项式环中的主要理想提出了一种新颖的结构理论。我们根据PDE，准时Hilbert方案，相对Weyl代数和连接构造来描述主要理想。从Ehrenpreis和Palamodov的意义上说，解决一个主要理想所描述的PDE等于计算Noetherian算子。我们为此任务开发了新算法，并提出了有效的实现方法。