Makar-Limanov's problem on values of polynomials on matrices

Abstract: Suppose $F$ is a field and $f(X_1, \dots,X_m)$ is a noncommutative polynomial. Makar-Limanov asked whether $f$ evaluated on $M_n(F)$ necessarily has some evaluation of bounded rank independent of $n.$ Answering this query, we show moreover that there are numbers $d\le \deg f$ and $m'\le m$ such that, under suitable restrictions on $F,$ for any $n \ge m'$ and any $\beta_i$ in $F$, there are matrices $A_1,\dots,A_m$ in~$M_n(F)$ such that $f(A_1,\dots,A_m)$ is diagonal, and the upper left $(n-m')\times (n-m')$ piece of $f(A_1,\dots,A_m)$ can be taken to be $\diag{\beta_1,\dots, \beta_{n-m'}}$, for indeterminates~$\beta_i$. When f is multilinear, $F$ can be an arbitrary field, and we can take $m' = m-1$. When $f$ is completely homogeneous, $F$ must be closed under $d$ roots. In general, $F$ can be any field closed under roots of polynomials of degree $\le d$. As Makar-Limanov observed, this leads to an immediate, characteristic-free proof of the Freiheitsatz for associative algebras over an algebraically closed field. Also, we show that if $f$ is not a polynomial identity of $k \times k$ matrices, then for generic matrices $Y_1,\dots,Y_m$, at least $n - k$ characteristic values of $f(Y_1,\dots,Y_m)$ are algebraically independent.