$A,B \in M(n,\mathbb C)$ with $AB=BA$ , then does there exist $P \in M(n,\mathbb C)$ and $f(X) , g(X) \in \mathbb C[X]$ such that $A=f(P), B=g(P)$?
Can commuting matrices $X,Y$ always be written as polynomials of some matrix $A$?