Let $k$ be an algebraically closed field of characteristic $p>0$, and let $(\mathfrak{g},[\cdot,\cdot],(\cdot)^{[p]})$ be a finite-dimensional restricted Lie algebra. Define the restricted nullcone of $\mathfrak{g}$ to be the set of all $a\in\mathfrak{g}$ with trivial restriction, ie:
$$\mathcal{N}_1(\mathfrak{g})=\{a\in\mathfrak{g}\mid a^{[p]}=0\}$$
We can consider $\mathfrak{g}$ to be a variety over $k$ with coordinate algebra $S(\mathfrak{g}^*)$, the symmetric algebra of the dual vector space of $\mathfrak{g}$. I often read in the literature that $\mathcal{N}_1(\mathfrak{g})\subset\mathfrak{g}$ is a subvariety, which means there is some way of interpreting the restriction map $[p]:\mathfrak{g}\to\mathfrak{g}$ as a "polynomial map," although I'm not quite sure what that means in this context. How is the condition $a^{[p]}=0$ a polynomial condition?
In short, how do we view the restriced nullcone as a subvariety of $\mathfrak{g}$? As a followup question, how do we see $k[\mathcal{N}_1(\mathfrak{g})]$ as a quotient of $S(\mathfrak{g}^*)$, that is, what is the radical ideal in $S(\mathfrak{g}^*)$ which defines $\mathcal{N}_1(\mathfrak{g})$?