A Log-Log Saving for Matrix-Algebra Length and Terseness

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 $\Mat_n(F)$. \v{S}itov 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} $\tau(n)$, i.e. the least upper bound for the length of words needed in Specht's theorem. Thus, for $n> 1$, $$\tau(n)\leq 4n\log_2 n-4n\log_2\log_2 n+10n+1.$$