Images of polynomial maps with constants

Abstract

Let $K$ be an algebraically closed field and $\mathrm{M}(2,K)$ be the $2\times 2$ matrix algebra over $K$ and $\mathrm{GL}(2,K)$ be the invertible elements in $\mathrm{M}(2,K)$. We explore the image of polynomials with constants, namely from the free algebra $\mathrm{M}(2,K)⟨x,y⟩$. In this article, we compute the images of the polynomial maps given by (a) generalized sum of powers $A{x}^{k_1}+B{y}^{k_2}$ and (b) generalized commutator map $Axy-Byx$, where $A$, $B$ are non-zero elements of $\mathrm{M}(2,K)$. We compute this in the first case by fixing a simultaneous conjugate pair for $A,B$ and it turns out that it is surjective in most of the cases. In the second case, we show that the image of the map is always a vector space.

Preliminary version; 34 pp;