TITLE: Variety consists of complex matrices whose power is $0$
QUESTION [6 upvotes]: Consider the set of $n\times n$ complex matrices $X\subseteq \mathbb{C}^{n^2}$ such that $X^n=0$. What is a minimal set of at most $n^2-1$ polynomials $f_1,\ldots,f_m$ such that the set of matrices is the variety of $f_1,\ldots,f_m$?
I don't really know where to start here. What polynomials would all those matrices be zero on (beside the obvious zero polynomial)?

REPLY [4 votes]: Hint: $X$ is a nilpotent matrix if and only if
$$
\operatorname{trace}(X^i) = 0
$$
For $i = 1,\dots,n$.