Images of polynomial maps with constants

Abstract

Let mathematical equation be the mathematical equation matrix algebra over mathematical equation and mathematical equation be the invertible elements in mathematical equation. Inspired by Kaplansky–Lv́ov conjecture, we explore the image of polynomials with constants, namely polynomials from the free algebra mathematical equation. In this article, we compute the images of the polynomial maps given by (a) generalized sum of powers mathematical equation and (b) generalized commutator map mathematical equation, where mathematical equation, mathematical equation are nonzero elements of mathematical equation when mathematical equation 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 mathematical equation, and it turns out that it is surjective in most cases.