关于群中匹配与域的向量子空间的结果与问题
Results and questions on matchings in groups and vector subspaces of fields
AI总结：
本文研究阿贝尔群与域扩张场景下的无环匹配问题，给出有限循环群无环匹配存在性结果，完成具线性无环匹配性质的域扩张分类，提出相关开放问题。
摘要：
从阿贝尔群的有限子集$A$到另一子集$B$的匹配是满足$a+f(a)$永远不属于$A$的双射$f:A\ ightarrow B$。若匹配由其重数函数唯一确定，则称其为无环匹配。受E. K. Wakeford关于对称张量典范形问题的启发，C. K. Fan与J. Losonczy在文献[16,26]中率先开展了阿贝尔群中匹配与无环匹配的研究，后续该研究被推广到域扩张中的向量子空间场景[13,1]。本文讨论了无环匹配性质与弱无环匹配性质，给出了有限循环群中无环匹配存在性的相关结果。在域扩张方面，我们对具有线性无环匹配性质的域扩张进行了完全分类。全文始终强调阿贝尔群中匹配与域扩张中匹配的类比关系，并提出了大量供后续研究的开放问题。
英文摘要：
A matching from a finite subset $A$ of an abelian group to another subset $B$ is a bijection $f:A\rightarrow B$ with the property that $a+f(a)$ never lies in $A$. A matching is called acyclic if it is uniquely determined by its multiplicity function. Motivated by a question of E. K. Wakeford on canonical forms for symmetric tensors, the study of matchings and acyclic matchings in abelian groups was initiated by C. K. Fan and J. Losonczy in [16, 26], and was later generalized to the context of vector subspaces in a field extension [13, 1]. We discuss the acyclic matching and weak acyclic matching properties and we provide results on the existence of acyclic matchings in finite cyclic groups. As for field extensions, we completely classify field extensions with the linear acyclic matching property. The analogy between matchings in abelian groups and in field extensions is highlighted throughout the paper and numerous open questions are presented for further inquiry.
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。
