On Homogeneous Polynomial Solutions of the Moisil-Théodoresco System in ℝ3

Abstract

The space of homogeneous polynomial solutions of degree k of the Moisil-Theodoresco system in ℝ3 is isomorphic to the real vector space MT+ (ℝ3; ℝ0,3 +; k) of homogeneous ℝ0,3 +-valued polynomial null-solutions of degree k of the Cauchy-Riemann operator D x in ℝ3. Hereby ℝ0,3 + is the even subalgebra of the Clifford algebra ℝ0,3. A structure theorem is proved for the elements W k ∈ MT+ (ℝ3; ℝ0,3 +; k), based essentially on conjugate harmonicity, and a general procedure is elaborated for constructing bases of MT+ (ℝ3; ℝ0,3 +; k).