UC Berkeley

Let K be a number field with ring of integers O. Consider the set of n x n alternating matrices with fixed rank, r

The principal ideas behind the proof were first introduced in a paper of Katznelson where the problem of counting matrices is reduced to one of counting lattice points. Because our matrices have entries in O rather than being restricted to the set of rational integers, the lattices of Katznelson are replaced in the present work with O-modules. The generalization from the rational numbers to K renders the standard tools of lattice-theory less directly applicable, and we rely on the Minkowski map and novel arguments to, at various turns, reduce to the lattice case, or abstract results from lattices to O-modules. Ancillary results in our work include a new formula for the discriminant of a torsion-free O-module in terms of its pseudo-basis and a novel structure theorem about the set of alternating matrices whose rows lie in a specified O-module.