Combinatorial Nullstellensatz and Tur\'an numbers of complete $r$-partite $r$-uniform hypergraphs

Abstract

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 $r \leq 4$.