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

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 $r \leq 4$.


Turán numbers of complete r-partite r-uniform hypergraphs
A hypergraph H = (V, E) consists of a set of vertices V and a set of edges E, each edge being some subset of V .A hypergraph is r-uniform if each edge in it contains exactly r vertices.An r-uniform hypergraph is r-partite if its set of vertices can be represented as a disjoint union of r parts with every edge containing one vertex from each part.The complete r-partite r-uniform hypergraph with parts of sizes s 1 , . . ., s r contains all s 1 • • • s r possible edges and is denoted by K (r) s1,...,sr .Let H be an r-uniform hypergraph.The Turán number ex(n, H) is the maximum number of edges in an r-uniform hypergraph on n vertices containing no copies of H.A classical result of Erdős [4] implies that for In [9], Mubayi conjectured that bound (1) is asymptotically tight.Recently, Pohoata and Zakharov [10] showed that this is true whenever s 1 , . . ., s r ≥ 2 and s r ≥ ((r − 1)(s extending earlier results of Alon, Kollár, Rónyai and Szabó [6,2] and Ma, Yuan and Zhang [8].
Nevertheless, the conjecture remains open even in a special case ex(n, K 2,...,2 ), which is often referred to as the Erdős box problem.The best known lower bound is due to Conlon, Pohoata and Zakharov [3], who showed that for any r ≥ 2, ex(n, (2)

Generalized Combinatorial Nullstellensatz
Let F be an arbitrary field, and let f ∈ F[x 1 , . . ., x r ] be a polynomial in r variables.A monomial Recall the famous Combinatorial Nullstellensatz by Alon (see Theorem 1.2 in [1]).
Theorem 1.1 (Alon, 1999).Let x d1 1 • • • x dr r be a monomial of f , and let any other monomial of f .Lasoń showed the following generalization of Combinatorial Nullstellensatz (see Theorem 2 in [7]).It should be mentioned that an even stronger theorem was proved by Schauz in 2008 (see Theorem 3.2(ii) in [12]).Theorem 1.2 (Lasoń, 2010).Let x d1 1 • • • x dr r be a maximal monomial of f .Then for any subsets A 1 , . . ., A r of F with sizes

Concluding remarks
The construction from Section 3 in the case r = 3 is structurally similar to the one given by Katz, Krop and Maggioni in [5].Their construction can be generalized to higher dimensions giving an alternative proof of Theorem 3.2 (private communication with Cosmin Pohoata; see also Proposition 11.2 in [13]).Our approach gives a simpler construction and a much shorter proof.
Motivated by the ideas discussed in Section 2, Rote posed a problem (see Problem 1 in [11]), equivalent to asking how large can the set Z(f ; B 1 , B 2 ) be for a polynomial of the form f (x, y) = xy + P (x) + Q(y) and sets B 1 , B 2 of size n each.Lemma 3.1 answers this question asymptotically if sets B 1 and B 2 are allowed to be taken from the finite field F p 2 .