有限域上的正定、半正定与全正矩阵
Positive definite, positive semidefinite and totally positive matrices over finite fields
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中文总结 AI 辅助
该研究针对有限域上的正定、半正定与全正矩阵，基于Cooper等人的有限域正元素定义给出不等价定义，结合Dwork定理推导计数公式并提出猜想。
中文摘要 AI 辅助
受实数域与复数域上正定（对应半正定）矩阵等价定义的启发，我们给出有限域上这类矩阵的四个（对应五个）不等价定义。我们的研究起点是Cooper-Hanna-Whitlatch（RMJ. Math., 2024）提出的有限域中正元素的最新定义，还利用该定义研究有限域上的全正矩阵。对所有这类情形，我们给出明确的计数公式或界；多数公式为新提出的，同时为完整性总结现有文献的结果。对5型半正定矩阵和全正矩阵，我们利用Weil ζ函数的有理性（即Dwork定理）给出结构公式，并猜想其为准多项式型公式。
英文摘要
Motivated by the equivalent definitions of positive definite (resp. positive semidefinite) matrices over real and complex fields, we give four (resp. five) inequivalent definitions for these matrices over finite fields. Our starting point is the recent definition due to Cooper--Hanna--Whitlatch (RMJ. Math., 2024) of positive elements in finite fields. We also use this definition to study totally positive matrices over finite fields. For all of these cases, we give explicit enumeration formulae or give bounds. Most of our formulas are new, but we summarize results from the existing literature for completeness. For positive semidefinite matrices of type 5 and totally positive matrices, we give structural formulas using the rationality of the Weil zeta function, i.e. Dwork's theorem, and conjecture a quasipolynomial-type formula.
