  In this note we describe how Laso\'n's generalization of Alon's Combinatorial
Nullstellensatz gives a framework for constructing lower bounds on the Tur\'an
number $\operatorname{ex}(n, K^{(r)}_{s_1,\dots,s_r})$ of the complete
$r$-partite $r$-uniform hypergraph $K^{(r)}_{s_1,\dots,s_r}$. To illustrate the
potential of this method, we give a short and simple explicit construction for
the Erd\H{o}s box problem, showing that $\operatorname{ex}(n,
K^{(r)}_{2,\dots,2}) = \Omega(n^{r - 1/r})$, which asymptotically matches best
known bounds when $r \leq 4$.
