Q:

Kernel of "polynomial evaluation map" for finite fields

For a field $k$ and $n$ a natural number (at least $1$) define the 'evaluation at a point' $k$-homomorphism
$$
\varphi_{k, n} : k[X_1, \ldots, X_n] \to k^{k^n} : f \mapsto (f(a))_{a \in k^n}.
$$
One easily shows by induction on $n$ that for $f \in k[X_1, \ldots, X_n]$ with $\deg_{X_i}(f) < \lvert k \rvert$ for all $i = 1, \ldots, n$ either $f = 0$ or $\varphi_{k, n}(f) \neq 0$. In particular it follows that $\varphi$ is injective if $k$ is infinite. If $k$ is the finite field with $q$ elements, it follows that the kernel of $\varphi_{k, 1}$ is the ideal generated by $X^q - X$.
Can a description of the kernel of $\varphi_{k, n}$ be given for finite fields $k$ and $n > 1$? Clearly it contains $X_i^q - X_i$ for all $i$, but is it the ideal generated by those polynomials?

A:

The kernel for $k=\Bbb F_q$ is the ideal $I$ generated by $X_i^q-X_i$ for all $i$.
Modulo $I$ each polynomial is congruent to one having degree $\le q-1$
in all variables. These polynomials form a vector space $V$ of dimension
$q^n$. Every map from $k^n$ to $k$ is represented by an element of $V$
and as $|V|=q^{q^n}$ then the only element of $V$ representing the
zero map is the zero polynomial. So the kernel of the "universal evaluation" is indeed $I$.

