On the images of generalized polynomials evaluated on matrices over an algebraically closed skew field

Abstract

Abstract Let K be a Makar-Limanov algebraically closed skew field. In the first part of this paper, we prove that the image of a generalized multilinear polynomial, with coefficients in K, evaluated over M m ( K ) , is M m ( K ) . In the second part, we show that any matrix in M m ( K ) may be written as the sum of three or fewer elements from the image of a generalized polynomial, with coefficients in K, evaluated over M m ( K ) .