Regular Functions of Several Quaternionic Variables and the Cauchy{fueter Complex

We employ a classical idea of Ehrenpreis, together with a new algebraic result, to give a new proof that regular functions of several quaternionic variables cannot have compact singularities. As a byproduct we characterize those inhomogeneous Cauchy{ Fueter systems which admit solutions on convex sets. Our method readily extends to the case of monogenic functions on Cliiord Algebras. We nally study a free resolution of the Cauchy{Fueter complex of diierential operators and we obtain some new duality theorems which hint at a hyperfunction theory of several quaternionic variables.