Images of polynomial maps with constants

Abstract

Let $$ be the $$ matrix algebra over $$ and $$ be the invertible elements in $$. Inspired by Kaplansky–Lv́ov conjecture, we explore the image of polynomials with constants, namely polynomials from the free algebra $$. In this article, we compute the images of the polynomial maps given by (a) generalized sum of powers $$ and (b) generalized commutator map $$, where $$, $$ are nonzero elements of $$ when $$ is an algebraically closed field. We show that the images of these maps are vector spaces. For the polynomial in (a), we compute the images by fixing a simultaneous conjugate pair for $$, and it turns out that it is surjective in most cases.