ac.commutative algebra - Jordan Form Over a Polynomial Ring

Let $X$ be the set of $ktimes k$ matrix with entries in $mathbb{C}$, and let $Min X$. The group $GL(k,mathbb{C})$ acts on $X$ by conjugation, and according to the Jordan decomposition theorem (see e.g wikipedia) somewhere in the orbit containing $M$ is a block diagonal matrix with non-zero entries only on the diagonal and superdiagonal.

Suppose now we consider $ktimes k$ matrices whose entries lie in the polynomial ring $mathbb{C}[z_{1},z_{2}, ldots ,z_{n}]$ and we study the action by conjugation of $GL(k,mathbb{C}[z_{1},z_{2}, ldots ,z_{n}])$. Then the Jordan decomposition theorem, as formulated above, clearly no longer holds. For example consider the matrix:

$ M=left(
{begin{array}{cc}
0 & 1 \
z^{p}_{1} & 0
end{array}}
right)$,

where in the above $p$ denotes a positive integer. If $p $ is odd, then $M$ cannot be diagonalized since the ring $mathbb{C}[z_{1},z_{2}, ldots ,z_{n}]$ does not contain the eigenvalues of $M$. On the other hand, if $p$ is even we still cannot diagonalize $M$ since when $z_{1}=0 $, $M$ is not diagonalizable.

My question is then what, if anything, remains of the Jordan decomposition in this case? Or equivalently given a $ktimes k$ matrix $M$ with entries in $mathbb{C}[z_{1},z_{2}, ldots ,z_{n}]$ are there any particularly simple matrices related to $M$ via conjugation by an element of $GL(k,mathbb{C}[z_{1},z_{2}, ldots ,z_{n}])$?