Surjectivity of maps induced on matrices by polynomials and entire functions

TL;DR

This paper establishes criteria for when polynomial and entire functions induce surjective maps on matrix algebras, using linear algebra and critical point analysis, extending to entire functions.

Contribution

It provides a necessary and sufficient condition for surjectivity of polynomial maps on matrix algebras and extends the result to entire functions.

Findings

Polynomial maps are surjective on matrices if and only if certain critical point conditions are met.

A similar criterion is established for entire functions.

The results are based on linear algebraic analysis of critical points.

Abstract

We determine a necessary and sufficient condition for a polynomial over an algebraically closed field $k$ to induce a surjective map on matrix algebras $M_n(k)$ for $n \ge 2$. The criterion is given in terms of critical points and uses simple linear algebra. Following that, we formulate and prove a corresponding result for entire functions as well.