Let $E$ be a finite dimensional $k$-vector space ($k$ may be assumed algebraically closed if needed) and $u,v \in \mathcal L(E)$ be two endomorphisms that commute ($u\circ v=v \circ u$).
Does there exist a third endomorphism $w$ and two polynomials $P,Q\in k[X]$ such that $u=P(w)$ and $v=Q(w)$?
I think the heart of the problem is to deal with the nilpotent subcase (where u and v are supposed to be both nilpotent). If $u$ (or $v$) is cyclic (ie only one Jordan block) this is classical, but I am not sure how to deal with the general case...
