If I have some $n \times n$ matrix $X$ (in my case, I happen to know that X is nilpotent and in Jordan normal form), how can I write the condition that $\text{rank}(X^r)= k$ as a polynomial equation (to represent a set of matrices as a variety) in terms of the entries of the matrix? I would be interested in a Sage program that does this, or writing my own program.
EDIT: I know that $\text{rank}(X) \leq k$ if and only if the determinant of every $(k+1) \times (k+1)$ minor of $X$ is zero, but I'm not sure how to get a set of polynomial equations which will give me exact equality.
The set of matrices with rank exactly $k$ is the open subset of those of rank at most $k$ defined by the non-vanishing of at least one $k$ by $k$ minor---thus, it is not naturally realized as a closed subvariety of the space of all matrices. You can use the usual trick to identify this with an affine variety if you wish: the point is that the set of non-zero elements $x$ of a field $F$ is isomorphic to the affine variety $xy=1$ in $F^2$.