PDE-preserving properties

Journal article, 2005

A continuous linear operator T, on the space of entire functions in d variables, is PDE-preserving for a given set ℙ ⊆ [ξ1, ..., ξd] of polynomials if it maps every kernel-set ker P(D), P ∈ ℙ, invariantly. It is clear that the set θ (ℙ) of PDE-preserving operators for P forms an algebra under composition. We study and link properties and structures on the operator side θ(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.