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

Abstract

Let ${\mathbb {F}}_q$ be a finite field of characteristic p and order q. The Chevalley–Warning Theorem asserts that the set V of common zeros of a collection of polynomials must satisfy $|V|\equiv 0\mod p$, provided the number of variables is sufficiently large with respect to the degrees of the polynomials. The Ax–Katz Theorem generalizes this by giving tight bounds for higher order p-divisibility for |V|. Besides the intrinsic algebraic interest of these results, they are also important tools in the Polynomial Method, particularly in the prime field case ${\mathbb {F}}_p$, where they have been used to prove many results in Combinatorial Number Theory. In this paper, we begin by explaining how arguments used by 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 Ax–Katz Theorem for ${\mathbb {F}}_p$, and thus also the Chevalley–Warning Theorem, 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,

$$\begin{aligned} V=\{{\textbf{a}}\in {\mathcal {B}}:\; f_1({{\textbf {a}}})\equiv 0\mod p^{m_1},\ldots ,f_s({{\textbf {a}}})\equiv 0\mod p^{m_s}\}, \end{aligned}$$

satisfies $|V|\equiv 0\mod p^m$, provided $n>(m-1)\max _{i\in [1,s]}\Big \{p^{m_i-1}\deg f_i\Big \}+ \sum \nolimits _{i=1}^{s}\frac{p^{m_i}-1}{p-1}\deg f_i.$ The introduction of the box ${\mathcal {B}}$ adds a degree of flexibility, in comparison to prior work of 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 ${\mathcal {I}}_j$, 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 by giving several examples, including a new proof of the exact value of the Davenport Constant ${\textsf{D}}(G)$ for finite abelian p-groups, and a streamlined proof of the Kemnitz Conjecture. We also derive some new results, for a finite abelian p-group G with exponent q, regarding the constant ${\textsf{s}}_{kq}(G)$, defined as the minimal integer $\ell$ such that any sequence of $\ell$ terms from G must contain a zero-sum subsequence of length kq. Among other results for this constant, we show that ${\textsf{s}}_{kq}(G)\le kq+{\textsf{D}}(G)-1$ provided $k>\frac{d(d-1)}{2}$ and $p> d(d-1)$, where , answering a problem of Xiaoyu He in the affirmative by removing all dependence on p from the bound for k.