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.
The classical Hartogs' Theorem  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  and then in .
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 . 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