Hartog's phenomenon for polyregular functions and projective dimension of related modules over a polynomial ring

Analysis of the module determining the properties of regular functions of several quaternionic variables

“…, n. It is easy to see that U i is a linear representation of the quarternion 1 ξ = x i0 +x i1 i+x i2 j +x i3 k. We denote by A n the submodule 1 Note added in proof: It should be noted that this representation differs slightly from the usual one which may be obtained from this one either by changing the sign of xi2 or by interchanging i and j. This does not effect any of the computations in this paper or in [3].…”

“…In this paper we will concentrate just on the algebraic properties of M n in (1) and we will continue the developements started in the earlier paper [3]. Our main motivation for this paper is to answer some of the questions posed in [13] and [15] concerning the syzygies of M n .…”

“…There are two main tools used in the proof of this result. The first is that, as seen in [3], we can write down an explicit Gröbner basis for A n for general n (with respect to the degree reverse lexicographical ordering on R). The second is that we can show that the (Castelnuovo) regularity of A n equals 2.…”

“…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 ≤ i ≤ n) denote 4n variables (n = 1, 2, .…”

“…. , n. If S n denotes a sheaf of generalized functions (e.g., S n is the sheaf of hyperfunctions or the sheaf of infinitely differentiable functions) and S P n denotes the left regular functions in S n , then analytic information concerning left regular functions is deduced from algebraic information concerning M n because of the isomorphism Hom R (M n , S n ) ∼ = S P n (see [1], [2], [3], [13], and [15]). …”