I am looking for an analogue of the Jordan normal form for nilpotent matrices over the polynomial ring ${\mathbb Z}[x_1, \dots, x_n]$. More precisely, is there a description for the orbits of action by conjugation of $GL_m({\mathbb Z}[x_1, \dots, x_n])$ on $M_{m \times m}({\mathbb Z}[x_1, \dots, x_n])$?

equivalentto $P$, instead of conjugated). This is the theory of elementary divisors. In your question, you keep conjugation, but your set of scalars is not a field, even not a P.I.D.! – Denis Serre Sep 21 '10 at 9:07