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ON COMPACT SINGULARITIES FOR REGULAR FUNCTIONS OF ONE QUATERNIONIC VARIABLE
 

Summary: ON COMPACT SINGULARITIES FOR REGULAR FUNCTIONS
OF ONE QUATERNIONIC VARIABLE
W.W. ADAMS, C.A. BERENSTEIN, P. LOUSTAUNAU, I. SABADINI, D.C. STRUPPA
Abstract. We prove that regular functions of one quaternionic variable which
satisfy a large class of differential equations cannot have compact singularities.
This result is equivalent to the fact that a large family of 8 \Theta 4 matrices has
torsion­free cokernel. The result (obvious in the complex case) easily extends
to Clifford algebras.
1. Introduction
The classical Hartogs' Theorem [6] on the removability of compact singularities
shows that a function of several complex variables cannot have compact singulari­
ties. This theorem was extended to the quaternionic case first in [9] and then in [1].
This last paper, in particular, shows how the quaternionic case can be considered in
the framework of the theory of overdetermined systems of partial differential equa­
tions developed in the sixties by Ehrenpreis [4, 5] and Palamodov [8]. In particular,
it is clear that the failure of Hartogs' Theorem in the one variable case (both in
the complex and in the quaternionic case) is due to the fact that both the Cauchy--
Riemann and the Cauchy--Fueter systems are not overdetermined when considered
in only one variable.
It is therefore natural to ask what happens when these one­variable systems are

  

Source: Adams, William W. - Department of Mathematics, University of Maryland at College Park

 

Collections: Mathematics