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arXiv · 2407.07486

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

Maarten De Boeck·Geertrui Van de Voorde
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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.

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Maarten De Boeck, Geertrui Van de Voorde. 2024-07-10. Anzahl theorems for trivially intersecting subspaces generating a non-singular subspace I: symplectic and hermitian forms. https://doi.org/10.1016/j.laa.2024.07.004

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