Treffer: A Generalization of the Chevalley-Warning and Ax-Katz Theorems with a View Towards Combinatorial Number Theory

We begin by explaining how arguments used by R. Wilson to give an elementary proof of the $\mathbb F_p$ case for the Ax-Katz Theorem can also be used to prove the following generalization of the Chevalley-Warning and Ax-Katz Theorems for $\mathbb F_p$, where we allow varying prime power moduli. Given any box $\mathcal B=\mathcal I_1\times\ldots\times\mathcal I_n$, with each $\mathcal I_j\subseteq\mathbb Z$ a complete system of residues modulo $p$, and a collection of nonzero polynomials $f_1,\ldots,f_s\in \mathbb Z[X_1,\ldots,X_n]$, then the set of common zeros inside the box, $$V=\{\mathbf a\in \mathcal B:\; f_1(\textbf a)\equiv 0\mod p^{m_1},\ldots,f_s(\textbf a)\equiv 0\mod p^{m_s}\},$$ satisfies $|V|\equiv 0\mod p^m$, provided $n>(m-1)\max_{i\in [1,s]}\Big\{p^{m_i-1}\mathsf{deg} f_i\Big\}+ \sum_{i=1}^{s}\frac{p^{m_i}-1}{p-1}\mathsf{deg} f_i.$ The introduction of the box $\mathcal B$ adds a degree of flexibility, in comparison to prior work of Zhi-Wei Sun. Indeed, incorporating the ideas of Sun, a weighted version of the above result is given. We continue by explaining how the added flexibility, combined with an appropriate use of Hensel's Lemma to choose the complete system of residues $\mathcal I_j$, effectively allows many combinatorial applications of the Chevalley-Warning and Ax-Katz Theorems, previously only valid for $\mathbb F_p^n$, to extend with bare minimal modification to validity for an arbitrary finite abelian $p$-group $G$. We illustrate this be giving several examples, including a new proof of the exact value of the Davenport Constant $\mathsf D(G)$ for finite abelian $p$-groups, a streamlined proof of the Kemnitz Conjecture, and the resolution of a problem of Xiaoyu He regarding zero-sums of length $k\exp(G)$ related to a conjecture of Kubertin.