Abstract

For a polynomial ring, R, in 4n variables over a field, we consider the submodule of R⁴ corresponding to the 4 × 4n matrix made up of n groupings of the linear representation of quarternions with variable entries (which corresponds to the Cauchy-Fueter operator in partial differential equations) and let M_n be the corresponding quotient module. We compute many homological properties of M_n 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 Gröbner bases.

Authors