Anzahl theorems for trivially intersecting subspaces generating a non-singular subspace I: symplectic and hermitian forms
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909262798651392 |
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| author | De Boeck, Maarten Van de Voorde, Geertrui |
| author_facet | De Boeck, Maarten Van de Voorde, Geertrui |
| contents | 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 $π$, we find the number of non-singular subspaces that are trivially intersecting with $π$ and span a non-singular subspace with $π$. Lower bounds for the quantity of such pairs where $π$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07486 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Anzahl theorems for trivially intersecting subspaces generating a non-singular subspace I: symplectic and hermitian forms De Boeck, Maarten Van de Voorde, Geertrui Combinatorics 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 $π$, we find the number of non-singular subspaces that are trivially intersecting with $π$ and span a non-singular subspace with $π$. Lower bounds for the quantity of such pairs where $π$ 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. |
| title | Anzahl theorems for trivially intersecting subspaces generating a non-singular subspace I: symplectic and hermitian forms |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.07486 |