Polynomial Maps with Constants on Matrix Algebra

TL;DR

This paper investigates the surjectivity of specific polynomial maps on matrix algebras, providing complete classifications for matrices of size 3 and 4 based on algebraic properties.

Contribution

It extends previous work by characterizing surjectivity conditions for polynomial maps with constants on matrix algebras of size 3 and 4, relating them to matrix nullity.

Findings

For n=2, surjectivity is fully characterized (from prior work).

For n=3,4, surjectivity depends on the nullity of A_2 and parameters n, k.

The paper provides necessary and sufficient conditions for surjectivity based on algebraic properties.

Abstract

Let $\mathcal A$ be an $\mathbb F$-algebra and $\omega \in \mathcal A\langle x_1, \ldots, x_m \rangle$ which defines a map $\mathcal A^m \rightarrow \mathcal A$ by evaluation, called a polynomial map with constant. We consider $\mathcal {A} = M_n(\mathbb{F})$, the algebra of $n \times n$ matrices over an algebraically closed field $\mathbb{F}$ of characteristic $0$, and polynomial maps given by $\omega(x_1, x_2) = A_1x_1^k + A_2x_2^k$, where $A_1,A_2\in M_n(\mathbb F)$. For $n=2$, the images of such a map is competely determined in an earlier work (Panja, S.; Saini, P.; Singh, A., Images of polynomial maps with constants, Mathematika 71 (2025), no. 3, Paper No. e70031). In this article, by assuming one of the coefficients, say $A_1$, is invertible, we relate the surjectivity of $\omega$ to the nullity of $A_2$. When $n=3, 4$, we completely classify the surjectivity of $\omega(x_1, x_2)$ by obtaining the necessary and sufficient condition in terms of $n$, $k$, and the nullity of $A_2$.