Pde - Preserving Properties

A continuous linear operator T , on the space of entire functions in d variables, is PDE-preserving for a given set P ⊆ C[ξ1, ..., ξd] of polynomials if it maps every kernel-set ker P (D), P ∈ P, invariantly. It is clear that the set O(P) of PDE-preserving operators for P forms an algebra under composition. We study and link properties and structures on the operator side O(P) versus the corresponding family P of polynomials. For our purposes, we introduce notions such as the PDE-preserving hull and basic sets for a given set P which, roughly, is the largest, respectively a minimal, collection of polynomials that generate all the PDE-preserving operators for P. We also describe PDE-preserving operators via a kernel theorem. We apply Hilbert’s Nullstellensatz.