Matrix evaluations of noncommutative rational functions and Waring problems

Let r be a nonconstant noncommutative rational function in m variables over an algebraically closed field K of characteristic 0. We show that for n large enough, there exists an X∈ M_n(K)^m such that r(X) has n distinct and nonzero eigenvalues. This result is used to study the linear and multiplicative Waring problems for matrix algebras. Concerning the linear problem, we show that for n large enough, every matrix in sl_n(K) can be written as r(Y)-r(Z) for some Y,Z∈ M_n(K)^m. We also discuss variations of this result for the case where r is a noncommutative polynomial. Concerning the multiplicative problem, we show, among other results, that if f and g are nonconstant polynomials, then, for n large enough, every nonscalar matrix in GL_n(K) can be written as f(Y)g(Z) for some Y,Z∈ M_n(K)^m.