Polynomial Maps with Constants on Matrix Algebra

Abstract

Let $\mathcal{A}$ be an $\mathbb{F}$-algebra and $\ddot{I}‰\in \mathcal{A}⟨x_1,\dots ,x_m⟩$ which defines a map ${\mathcal{A}}^m\to \mathcal{A}$ 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 $\ddot{I}‰(x_1,x_2)=A_1{x}_1^k+A_2{x}_2^k$, 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 $\ddot{I}‰$ to the nullity of $A_2$. When $n=3,4$, we completely classify the surjectivity of $\ddot{I}‰(x_1,x_2)$ by obtaining the necessary and sufficient condition in terms of $n$, $k$, and the nullity of $A_2$.

Preliminary Version; 22 pages