Surjectivity of Maps Induced on Matrices by Polynomials and Entire Functions

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