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

Abstract

Suppose $F$ is an infinite field and let $f \in F\{X_1, \dots,X_m\}$ be a noncommutative polynomial. Partially answering a query of Makar-Limanov, we show that there are numbers $d$ and $m'$ such that, if $F$ is closed under taking $d$th roots, for any $n \ge m'$ there are matrices $A_1,\dots,A_m$ in~$M_n(F)$ such that $f(A_1,\dots,A_m)$ is upper triangular with $n-m'$ prescribed diagonal entries. When f is homogeneous, $f(A_1,\dots,A_m)$ is diagonal with $n-m'$ prescribed diagonal entries. When f is multilinear, we can take $d=1$ and $m' = [\frac{m-1}{2}]$, 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 $\beta_i$. Furthermore, if $f$ is not a polynomial identity of $ k \times k $ matrices, then at least $ n - k $ characteristic values of $ f(A_1,\dots,A_m) $ may be taken to be algebraically independent.