Left Regular Polynomials in Even Dimensions, and Tensor Products of Clifford Algebras

Abstract

In [4] Fueter describes a method for characterizing homogeneous polynomial solutions to the quaternionic analogue of the Cauchy-Riemann equations. In [3] Delanghe demonstrates that this method may be generalized to characterize homogeneous polynomial solutions to a generalized Cauchy-Riemann equation defined over a Clifford algebra. The function theory associated with this particular equation has been extensively pursued in recent years by a number of authors (eg [2,5,7,8,9,10]). However, the author shows in [7] that the methods used in [3,4] may be extended to characterize homogeneous polynomial solutions to homogeneous, first order, constant coefficient differential equations defined over arbitrary, finite dimensional, associative algebras with identity. For this reason it would appear desirable to find another method to characterize these solutions, to generalized Cauchy-Riemann equations over Clifford algebras, which is more closely aligned to the properties of Clifford algebras.