Primary Ideals and Their Differential Equations

Abstract

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. Keywords primary ideals • linear partial differential equations • Noetherian operators • differential operators • punctual Hilbert scheme • Weyl algebra • join of ideals • symbolic powers. Mathematics Subject Classification (2010) 13N10 • 35E05 • 13N99 • 13A15.

Linear PDE with constant coefficients

“…The output is the description of Sol(M) sought in (2.7). That description is unique up to basis change, in the sense of [12,Remark 3.8], by the discussion in Section 4. Our method is implemented in a Macaulay2 command, called solvePDE and to be described in Section 5.…”

“…Other references with different emphases include [5,25,29]. For a perspective from commutative algebra see [11,12].…”

“…is an invariant of the module M, denoted amult(M), and known as the arithmetic multiplicity of M. These numbers can be computed in Macaulay2 as in [12,Remark 5.1]. We return to these invariants in Theorem 4.…”

“…We present techniques that are based on the Fundamental Principle of Ehrenpreis and Palamodov, as discussed in the classical books [7,17,23,32]. We utilize the theory of differential primary decomposition [12]. While deriving (1.5) from (1.…”

“…Theorem 3.8 characterizes PDE whose solution space is finite dimensional. Section 4 features the theory of differential primary decomposition [9,12]. Theorem 4.4 shows how this theory yields the integral representations promised by Ehrenpreis- Palamodov. This result appeared implicitly in the analysis literature, but the present algebraic form is new.…”