Surjectivity of polynomial maps on Matrices

TL;DR

This paper investigates the surjectivity of polynomial maps on matrix algebras over various fields, establishing conditions under which these maps are onto, with specific results for complex, finite, real, and quaternionic matrices.

Contribution

It provides new criteria for the surjectivity of diagonal polynomial maps on matrix algebras over different fields, including real, complex, finite, and quaternionic matrices.

Findings

Surjective for complex matrices when m≥2.

Surjective for large finite fields when m≥2.

Surjective for quaternionic matrices when m≥2.

Abstract

For $n\geq 2$, we consider the map on $M_n(\mathbb K)$ given by evaluation of a polynomial $f(X_1, \ldots, X_m)$ over the field $\mathbb K$. In this article, we explore the image of the diagonal map given by $f=\delta_1 X_1^{k_1} + \delta_2 X_2^{k_2} + \cdots +\delta_m X_m^{k_m}$ in terms of the solution of certain equations over $\mathbb K$. In particular, we show that for $m\geq 2$, 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 $\delta _1\delta_2>0$ (in that case the image misses negative scalars), and the map is surjective for $m\geq 3$. We further show that on $M_n(\mathbb H)$ the diagonal map is surjective for $m\geq 2$, where $\mathbb H$ is the algebra of Hamiltonian quaternions.