论文标题
对雪佛兰 - 巡洋舰和斧头定理的概括，朝向组合数理论
A Generalization of the Chevalley-Warning and Ax-Katz Theorems with a View Towards Combinatorial Number Theory
论文作者
论文摘要
首先，我们解释了R. Wilson使用的参数如何为AX-KATZ定理提供$ \ Mathbb f_p $ case的基本证明，也可以用于证明Chevalley-Warning和ax-Katz定理的以下概括，用于$ \ Mathbb f_p $，在我们允许VARY PRINE PRINE PORIDE PORIDE模块化。给定的任何框$ \ MATHCAL B = \ MATHCAL I_1 \ times \ ldots \ times \ times \ times \ Mathcal i_n $，每个$ \ Mathcal I_J \ subseteq \ subseteq \ Mathbb z $一个完整的残留物modulo $ p $和非零polynomials $ f_1，\ ldots， z [x_1，\ ldots，x_n] $，然后在框内，$$ v = \ {\ mathbf a \ in \ mathcal b：\;;; f_1（\ textbf a）\ equiv 0 \ mod p^{m_1}，\ ldots，f_s（\ textbf a）\ equiv 0 \ equiv 0 \ mod p^{m_s} \}，$ | [1,s]}\Big\{p^{m_i-1}\mathsf{deg} f_i\Big\}+ \sum_{i=1}^{s}\frac{p^{m_i}-1}{p-1}\mathsf{deg} f_i.$ The introduction of the box $\mathcal B$ adds a与Zhi-Wei Sun的先前工作相比，灵活性程度。实际上，结合了太阳的想法，给出了上述结果的加权版本。 We continue by explaining how the added flexibility, combined with an appropriate use of Hensel's Lemma to choose the complete system of residues $\mathcal I_j$, effectively allows many combinatorial applications of the Chevalley-Warning and Ax-Katz Theorems, previously only valid for $\mathbb F_p^n$, to extend with bare minimal modification to validity for an arbitrary finite abelian $p$-group $ g $。我们说明了这一点，包括有限的Abelian $ p $ -groups的Davenport常量$ \ mathsf d（g）$的确切价值的新证明，这是Kemnitz猜想的简化证明，以及针对Xiaoyu He of Xiaoyu He of thenge of thengue $ k \ exp（g）的零sums $ kuectiur kuert kuert ku ku ku的零s。
We begin by explaining how arguments used by R. Wilson to give an elementary proof of the $\mathbb F_p$ case for the Ax-Katz Theorem can also be used to prove the following generalization of the Chevalley-Warning and Ax-Katz Theorems for $\mathbb F_p$, where we allow varying prime power moduli. Given any box $\mathcal B=\mathcal I_1\times\ldots\times\mathcal I_n$, with each $\mathcal I_j\subseteq\mathbb Z$ a complete system of residues modulo $p$, and a collection of nonzero polynomials $f_1,\ldots,f_s\in \mathbb Z[X_1,\ldots,X_n]$, then the set of common zeros inside the box, $$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.
