平凡相交子空间生成非奇异子空间的Anzahl定理 I：辛型和厄米型
Anzahl theorems for trivially intersecting subspaces generating a non-singular subspace I: symplectic and hermitian forms
AI总结：
本文针对辛型和厄米型非退化形式，推导出与给定子空间平凡相交且共同生成非奇异子空间的非奇异子空间数量的显式公式，从而给出精确比例并改进已知下界。
摘要：
本文解决了定义在向量空间上的辛型和厄米型非退化形式的经典计数问题：给定一个子空间 $\pi$，我们求出与 $\pi$ 平凡相交且与 $\pi$ 张成非奇异子空间的非奇异子空间的数量。当 $\pi$ 为非奇异时，此类对的数量下界首先在“Glasby, Niemeyer, Praeger (Finite Fields Appl., 2022)”中研究，随后在“Glasby, Ihringer, Mattheus (Des. Codes Cryptogr., 2023)”中得到改进，并在“Glasby, Niemeyer, Praeger (Linear Algebra Appl., 2022)”中被推广。在本文中，我们推导出显式公式，从而能够给出精确比例并改进已知下界。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
