发表机构
University of Delaware; Université Laval; University of Regina; Hong Kong University of Science and Technology（特拉华大学; 拉瓦尔大学; 里贾纳大学; 香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了有限域上逐元素保正算子分类中$n=2$、$q\boldsymbol{\text{≡}}1\boldsymbol{\text{(mod}\text{ }4\text{)}}$且$q$非平方数的遗留情形，结合幂等归约与二次特征性质完成了所有有限域及固定维数的完整分类。
AI 中文摘要
我们称有限域$\boldsymbol{\text{F}}_q$上的矩阵为正定的，若它是对称矩阵且其所有顺序主子式均为$\boldsymbol{\text{F}}_q$中的非零平方元。在作者此前的工作[《代数学杂志》，2025年]中，已对所有$n\boldsymbol{\text{≥}}2$时$M_n(\boldsymbol{\text{F}}_q)$上的逐元素保正算子进行了分类，仅余一个待解决的情形：$n=2$、$q\boldsymbol{\text{≡}}1\boldsymbol{\text{(mod}\text{ }4\text{)}}$且$q$非平方数。我们通过证明当$q\boldsymbol{\text{≡}}1\boldsymbol{\text{(mod}\text{ }4\text{)}}$时，$M_2(\boldsymbol{\text{F}}_q)$上的每一个保正算子都在非零平方元集合$\boldsymbol{\text{F}}_q^+$上是单射，从而解决了该情形。证明过程将保正算子的幂等归约与二次特征的已知性质相结合。这一结果完成了所有有限域及所有固定维数上逐元素保正算子的完整分类。
英文摘要
We say that a matrix over a finite field $\mathbb{F}_q$ is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in $\mathbb{F}_q$. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on $M_n(\mathbb{F}_q)$ were classified in every case except when $n=2$, $q\equiv1\pmod4$, and $q$ is not a square. We settle this remaining case, thereby completing the classification of entrywise positivity preservers over every finite field and in every dimension $n\ge2$. Our proof is based on a novel idempotent reduction that not only resolves the remaining case but also yields a self-contained proof of the complete classification, while avoiding several technical results used in the earlier arguments. As a further application of the same reduction, we classify the entrywise preservers of strongly nonsingular matrices, i.e., matrices whose leading principal minors are all nonzero. We also prove a more general theorem in odd characteristic: for every prescribed sign pattern of nonzero leading principal minors of matrices of a fixed dimension $n\ge2$, the entrywise preservers are precisely the positive scalar multiples of field automorphisms. Thus, in odd characteristic, preserving any nonzero leading-principal-minor sign pattern surprisingly forces the preservation of every such sign pattern.
Comments13 pages; latex; major update
