Polynomial maps with constants on matrix algebra

Abstract

Let A be an F-algebra and ω∈A〈x1,…,xm〉 which defines a map Am→A by evaluation, called a polynomial map with constant. We consider A=Mn(F), the algebra of n×n matrices over an algebraically closed field F of characteristic 0, and polynomial maps given by ω(x1,x2)=A1x1k+A2x2k, where A1,A2∈Mn(F). For n=2, the images of such maps are completely determined in an earlier work (Panja et al., 2025 [24]). In this article, by assuming one of the coefficients, say A1, is invertible, we relate the surjectivity of ω to the nullity of A2. When n=3,4, we completely classify the surjectivity of ω(x1,x2) by obtaining the necessary and sufficient condition in terms of n, k, and the nullity of A2.