Combinatorial Theory

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.

Mathematics Subject Classifications: 51A50, 51E20

Keywords: Quadratic forms, counting, non-singular subspace