Surjectivity of polynomial maps on Matrices

Abstract

For $n\ge 2$, we consider the map on $M_n(\mathbb{K})$ given by evaluation of a polynomial $f(X_1,\dots ,X_m)$ over the field $\mathbb{K}$. In this article, we explore the image of the diagonal map given by $f=\hat{I}{´}_1{X}_1^{k_1}+\hat{I}{´}_2{X}_2^{k_2}+\cdots +\hat{I}{´}_m{X}_m^{k_m}$ in terms of the solution of certain equations over $\mathbb{K}$. In particular, we show that for $m\ge 2$, the diagonal map is surjective when (a) $\mathbb{K}=\mathbb{C}$, (b) $\mathbb{K}={\mathbb{F}}_q$ for large enough $q$. Moreover, when $\mathbb{K}=\mathbb{R}$ and $m=2$ it is surjective except when $n$ is odd, $k_1,k_2$ are both even, and $\hat{I}{´}_1\hat{I}{´}_2>0$ (in that case the image misses negative scalars), and the map is surjective for $m\ge 3$. We further show that on $M_n(\mathbb{H})$ the diagonal map is surjective for $m\ge 2$, where $\mathbb{H}$ is the algebra of Hamiltonian quaternions.