 
Summary: ANALYSIS OF THE MODULE DETERMINING THE
PROPERTIES OF REGULAR FUNCTIONS OF SEVERAL
QUATERNIONIC VARIABLES
WILLIAM W. ADAMS AND PHILIPPE LOUSTAUNAU
Abstract. For a polynomial ring, R, in 4n variables over a field, we consider
the submodule of R 4 corresponding to the 4 \Theta 4n matrix made up of n group
ings of the linear representation of quarternions with variable entries (which
corresponds to the CauchyFueter operator in partial differential equations)
and let Mn be the corresponding quotient module. We compute many homo
logical properties of Mn including the degrees of all of its syzygies, as well as
its Betti numbers, Hilbert function, and dimension. We give similar results
for its leading term module with respect to the degree reverse lexicographical
ordering. The basic tool in the paper is the theory of Gr¨obner bases.
1. Introduction
In several recent papers, [1], [2] and [3], the authors and their colleagues have
applied to certain analytic questions some algebraic properties of the following
module. Let x i0 ; x i1 ; x i2 ; x i3 (1 Ÿ i Ÿ n) denote 4n variables (n = 1; 2; : : : ), let k
be any field, and let R = k[x i0 ; x i1 ; x i2 ; x i3 j1 Ÿ i Ÿ n] denote the corresponding
polynomial ring in the given 4n variables (in the analytic applications k = C , but
the specific field plays no role in the current paper). We consider the 4 \Theta 4n matrix
