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{
    "pk": 64990,
    "title": "Anzahl theorems for disjoint subspaces generating a non-degenerate subspace: quadratic forms",
    "subtitle": null,
    "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 nonsingular subspace with \\(\\pi\\). Lower bounds for the quantity of such pairs where \\(\\pi\\) is nonsingular were first studied in [S. P. Glasby, Alice C. Niemeyer, and Cheryl E. Praeger. The probability of spanning a classical space by two non-degenerate subspaces of complementary dimensions. Finite Fields Appl., 82:31, 2022], which was later improved for even-dimensional subspaces in [S. P. Glasby, F. Ihringer, and S. Mattheus. The proportion of non-degenerate complementary subspaces in classical spaces. Des. Codes Cryptography, 91(9):2879– 2891, 2023] and generalised in [S.P. Glasby, A.C. Niemeyer, and C.E. Praeger. Random generation of direct sums of finite non-degenerate subspaces. Linear Algebra Appl., 649:408–432, 2022]. The explicit formulae, which give the exact proportion and improve the known lower bounds were derived in the symplectic and Hermitian case in [M. De Boeck and G. Van de Voorde. Anzahl theorems for trivially intersecting subspaces generating a non-singular subspace. I: Symplectic and Hermitian forms. Linear Algebra and its Applications, 699:367–402, 2024]. This paper deals with the more complicated quadratic case.\n \nMathematics Subject Classifications: 51A50, 51E20\n \nKeywords: Quadratic forms, counting, non-singular subspace",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Quadratic forms"
        },
        {
            "word": "counting"
        },
        {
            "word": "non-singular subspace"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/1324k25d",
    "frozenauthors": [
        {
            "first_name": "Maarten",
            "middle_name": "",
            "last_name": "De Boeck",
            "name_suffix": "",
            "institution": "Stichting Leerplanontwikkeling, Amersfoort, Netherlands -- Department of Mathematics, Computer Science and Statistics, Ghent University, Gent, Flanders, Belgium",
            "department": ""
        },
        {
            "first_name": "Geertrui",
            "middle_name": "",
            "last_name": "Van de Voorde",
            "name_suffix": "",
            "institution": "School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, 8140 Christchurch, New Zealand",
            "department": ""
        }
    ],
    "date_submitted": "2025-07-15T15:24:39Z",
    "date_accepted": "2025-07-15T15:24:39Z",
    "date_published": "2025-07-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64990/galley/49800/download/"
        }
    ]
}