AI 中文总结
本文研究矩阵代数长度，在已有估计基础上获得对数-对数节省，证明了$\ell(\Mat_n(F))$新的上界。利用Specht定理和迹论证，得出简洁性$\tau(n)$的上界，为相关研究提供了新的理论结果。
AI 中文摘要
设$\ell(\Mat_n(F))$表示域$F$上全矩阵代数的长度，即跨越$\Mat_n(F)$所需的最小字长在$\Mat_n(F)$的所有生成集$S$上的最大值。Šitov证明了一般估计$\ell(\Mat_n(F)) \leq 2n\log_2 n+4n-4$。本文旨在获得对数-对数节省，并证明对于每个$n>1$，$\ell(\Mat_n(F)) \leq 2n\log_2 n-2n\log_2\log_2 n+5n$。Specht定理给出了复$n\times n$矩阵酉相似性的字准则。Freedman--Gupta--Guralnick的迹论证表明，$\ell(\Mat_n(F))$的任何上界都可用于界定简洁性$\tau(n)$。因此，对于$n>1$，$\tau(n)\leq 4n\log_2 n-4n\log_2\log_2 n+10n+1$。
英文摘要
Let $\ell(\Mat_n(F))$ denote the length of the full matrix algebra for a field $F$, i.e. the largest of the least word length needed to span $\Mat_n(F)$, over all generating sets $S$ of $\Mat_n(F)$. Šitov proved the general estimate $$ \ell(\Mat_n(F)) \leq 2n\log_2 n+4n-4. $$ The purpose of this paper is to obtain a log-log saving, and prove that for every $n>1$, $$ \ell(\Mat_n(F)) \leq 2n\log_2 n-2n\log_2\log_2 n+5n. $$ A theorem of Specht gives a word-criterion for unitary similarity of complex $n\times n$ matrices. The trace argument of Freedman--Gupta--Guralnick, as used by Pappacena, shows that any upper bound on $\ell(\Mat_n(F))$ can be used to bound the \emph{terseness} $τ(n)$, i.e. the least upper bound for the length of words needed in Specht's theorem. Thus, for $n> 1$, $$ τ(n)\leq 4n\log_2 n-4n\log_2\log_2 n+10n+1. $$
Comments7 pages+ references. Incorporated several improvements, and added a new reference. If you have any comments, please let me know
