Anzahl theorems for trivially intersecting subspaces generating a non-singular subspace I: symplectic and hermitian forms

TL;DR

This paper provides explicit formulas for counting non-singular subspaces intersecting trivially with a given subspace in symplectic and hermitian spaces, improving previous bounds and offering exact proportions.

Contribution

It derives explicit formulas for counting such subspaces, advancing the understanding of their distribution in symplectic and hermitian spaces.

Findings

Exact formulas for counting non-singular subspaces

Improved lower bounds for the quantity of such pairs

Determination of the exact proportion of these subspaces

Abstract

In this paper, we solve a classical counting problem for non-degenerate forms of symplectic and hermitian type defined on a vector space: given a subspace $\pi$, we find the number of non-singular subspaces that are trivially intersecting with $\pi$ and span a non-singular subspace with $\pi$. Lower bounds for the quantity of such pairs where $\pi$ is non-singular were first studied in ``Glasby, Niemeyer, Praeger (Finite Fields Appl., 2022)'', which was later improved in ``Glasby, Ihringer, Mattheus (Des. Codes Cryptogr., 2023)'' and generalised in ``Glasby, Niemeyer, Praeger (Linear Algebra Appl., 2022)''. In this paper, we derive explicit formulae, which allow us to give the exact proportion and improve the known lower bounds.