 Solution

Write each hyperplane as $h_j(x)=0$, normalized so that $h_j(0)=1$, and putThen $P(0)=1$ and $P$ vanishes at every other point of ${\mathbb{F}}_q^n$. Reduce $P$ modulo $x_i^q-x_i$ in every variable. This preserves its function on ${\mathbb{F}}_q^n$, does not increase total degree, and gives the unique representative with each variable degree at most $q-1$.

The unique reduced polynomial for the delta function at zero iswhose total degree is $(q-1)n$. HenceUniqueness follows equally from the Alon-Tarsi lemma on the grids $A_i={\mathbb{F}}_q$.