有限域上的奇异Cholesky纤维

中文摘要 AI 辅助

对于有限域$\boldsymbol{\text{F}}_q$，考虑从上三角矩阵到对称矩阵的三角Cholesky映射$\boldsymbol{\text{Γ}}_{n,q}(U)=U^TU$。广义Cholesky理论描述了所有顺序主子式均非零的正则轨迹，但未确定奇异目标上方纤维的重数或根秩。我们针对该奇异边界发展了固定目标纤维理论，解答了Cooper与Whitlatch提出的三个问题。主要结果围绕零纤维展开：在$\boldsymbol{\text{F}}_2$上，我们构造了平方零上三角矩阵与满足$U^TU=0$的上三角矩阵之间显式、可逆、保秩的递归双射；在每个有限域上，我们确定了该零纤维的完整秩分布：偶特征下，平方零递归（及由此产生的秩细化等势性）在每个$\boldsymbol{\text{F}}_{2^e}$上均成立，而奇特征下，其失效由显式二次特征修正项衡量。我们通过证明任意对称目标上方纤维基数的精确首元递归，将这些结果置于统一框架中：其正则特化给出顺序主子式锥上的常纤维大小，而零元分支解释了新的奇异行为。针对二元对角目标，我们进一步将秩细化计数压缩至$O(n^2)$次整数算术转换，并证明其依赖于对角元的顺序而非仅数量。附录中收集了可执行实现与详尽的低阶验证。

英文摘要

For a finite field $\F_q$, consider the triangular Cholesky map $Γ_{n,q}(U)=U^TU$ from upper triangular matrices to symmetric matrices. Generalized Cholesky theory describes the regular locus on which all leading principal minors are nonzero, but it does not determine the multiplicities or root ranks in a fiber over a singular target. We develop a fixed-target fiber theory for this singular boundary and answer three questions posed by Cooper and Whitlatch. Our principal results concern the zero fiber. Over $\F_2$ we construct an explicit, invertible, rank-preserving recursive bijection between square-zero upper triangular matrices and upper triangular matrices satisfying $U^TU=0$. Over every finite field we determine the entire rank distribution of this zero fiber: in even characteristic the square-zero recurrence, and hence the rank-refined equinumerosity, persists over every $\F_{2^e}$, whereas in odd characteristic its failure is measured by an explicit quadratic-character correction. We place these results in a uniform framework by proving an exact first-pivot recursion for the fiber cardinality over an arbitrary symmetric target. Its regular specialization gives the constant fiber sizes on leading-principal-minor cones, while its zero-pivot branch explains the new singular behavior. For binary diagonal targets we further compress the rank-refined count to $O(n^2)$ integer-arithmetic transitions and show that it depends on the order, not merely the number, of the diagonal entries. Executable implementations and exhaustive low-order checks are collected in an appendix.

发表机构

• Suzhou University of Technology（苏州科技大学）
• North China University of Technology（华北理工大学）

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