The Nullstellensatz for systems of PDE

Given an ideal I in A, the polynomial ring in n-indeterminates, the affine variety of I is the set of common zeros in C-n of all the polynomials that belong to I, and the Hilbert Nullstellensatz states that there is a bijecti ve correspondence between these affine varieties and radical ideals of A. I f, on the other hand, one thinks of a polynomial as a (constant coefficient ) partial differential operator, then instead of its zeros in C-n, one can consider its zeros, i.e., its homogeneous solutions, in various function an d distribution spaces. An advantage of this point of view is that one can t hen consider not only the zeros of ideals of A, but also the zeros of submo dules of free modules over A (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 solution s of PDEs are being located, and this paper considers the case of the class ical spaces. This question is related to the more general question of embed ding a partial differential st;stem in a (two-sided) complex with minimal h omology, This paper also explains dow these questions are related to some q uestions in control theory. (C) 1999 Academic Press.