Anzahl theorems for disjoint subspaces generating a non-degenerate subspace II: quadratic forms

TL;DR

This paper provides explicit formulas for counting non-singular subspaces intersecting trivially with a given subspace in quadratic forms over finite fields, extending previous work on symplectic and Hermitian cases.

Contribution

It derives the first explicit formulas for counting such subspaces in the quadratic form case, improving bounds and understanding in this more complex setting.

Findings

Derived explicit formulas for quadratic forms case

Improved bounds on the number of such subspaces

Extended previous results from symplectic and Hermitian cases

Abstract

In this paper, we solve a classical counting problem for non-degenerate quadratic forms defined on a vector space in odd characteristic; given a subspace $\pi$, we determine 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 for even-dimensional subspaces in `Glasby, Ihringer, Mattheus (Des. Codes Cryptogr., 2023)' and generalised in `Glasby, Niemeyer, Praeger (Linear Algebra Appl., 2022)'. The explicit formulae, which allow us to give the exact proportion and improve the known lower bounds were derived in the symplectic and Hermitian case in `De Boeck and Van de Voorde (Linear Algebra Appl. 2024)'. This paper deals with the more complicated quadratic case.