On the Mesyan conjecture

The Waring problem for matrix algebras, II

“…It is a standard fact that if $$p$$ is a positive integer such that $$m<2p$$, then $$f$$ cannot be a polynomial identity of $$M_p(F)$$. Using this fact, it can be easily shown that under the same assumption $$f$$ also cannot be a central polynomial for $$M_p(F)$$ (see, e.g., [3, Corollary 2.4]). (II)Let $$F$$ be an algebraically closed field of characteristic 0, let $$p$$ be a prime, and let $$f$$ be a polynomial that is neither a polynomial identity nor a central polynomial of $$M_p(F)$$. Then $$f(M_p(F))$$ contains a diagonal matrix $$D$$ with distinct eigenvalues on the diagonal.…”