The Nullstellensatz for Systems of PDE

Given an ideal I in , the polynomial ring in n-indeterminates, the affine variety of I is the set of common zeros in ⁿ of all the polynomials that belong to I, and the Hilbert Nullstellensatz states that there is a bijective correspondence between these affine varieties and radical ideals of . If, on the other hand, one thinks of a polynomial as a (constant coefficient) partial differential operator, then instead of its zeros in ⁿ, one can consider its zeros, i.e., its homogeneous solutions, in various function and distribution spaces. An advantage of this point of view is that one can then consider not only the zeros of ideals of , but also the zeros of submodules of free modules over (i.e., of systems of PDEs). The question then arises as to what is the analogue here of the Hilbert Nullstellensatz. The answer clearly depends on the function–distribution space in which solutions of PDEs are being located, and this paper considers the case of the classical spaces. This question is related to the more general question of embedding a partial differential system in a (two-sided) complex with minimal homology. This paper also explains how these questions are related to some questions in control theory.