The hyperdeterminant $D(A)$ is a multidimensional generalization of the determinant. It is a polynomial in the entries of a $(k_1+1)\times (k_2+1)\times\cdots \times (k_n+1)$ array $A$. The hyperdeterminant is defined when $2\max k_i\leq \sum k_i$. For further information see for instance http://arxiv.org/pdf/1301.0472v1.pdf.
Assuming that $(k_1,\dots,k_n)$ satisfies the above condition, let $N(q)$ be the number of nonzero $(k_1+1)\times\cdots\times(k_n+1)$ hyperdeterminants over the finite field $\mathbb{F}_q$. For instance, if $n=2$ then $k_1=k_2$, and the number of $m\times m$ matrices over $\mathbb{F}_q$ with nonzero determinant is well-known and easily seen to equal $(q^m-1)(q^m-q)\cdots (q^m-q^{m-1})$. A few years ago some M.I.T. graduate students computed that for $2\times 2\times 2$ hyperdeterminants we get a polynomial in $q$, though I seem to have misplaced the formula. Can anything be said about the general case? Do we always get a polynomial in $q$? (This seems unlikely to me.) Can someone compute the $2\times 2\times 3$ case?