Surjectivity of polynomial maps on matrices

Abstract

For n⩾2, we consider the polynomial maps on Mn(K) given by evaluation of a polynomial f(X1,…,Xm) over the field K. We explore the image of the diagonal map given by in terms of the solution of certain equations over K. We show that when K=R and m=2, it is surjective except when n is odd, δ1δ2>0, and k1,k2 are both even (in that case, the image misses negative scalars), and the map is surjective for m⩾3. We further show that on Mn(H) (even with H coefficients) the diagonal map is surjective for m⩾2, where H is the algebra of Hamilton’s quaternions.