2022/08/26 by David J. Grynkiewicz, Grynkiewicz, David J.
Mathematics · Computer Science · #Analytic Number Theory Research #Coding theory and cryptography #Cryptography and Residue Arithmetic
paper · pdf · doi:10.48550/arxiv.2208.12895
We begin by explaining how arguments used by R. Wilson to give an elementary proof of the \mathbb Fp 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 Fp, where we allow varying prime power moduli. Given any box \mathcal B=\mathcal I1×…×\mathcal In, with each \mathcal Ij⊆\mathbb Z a complete system of residues modulo p, and a collection of nonzero polynomials f1,…,fs∈ \mathbb Z[X1,…,Xn], then the set of common zeros inside the box, V=\\mathbf a∈ \mathcal B: f1(\textbf a)≡ 0\mod pm1,…,fs(\textbf a)≡ 0\mod pms\, satisfies |V|≡ 0\mod pm, provided n>(m-1)maxi∈ [1,s]\pmi-1deg fi\+ ∑i=1s\fracpmi-1p-1deg fi. 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 Ij, effectively allows many combinatorial applications of the Chevalley-Warning and Ax-Katz Theorems, previously only valid for \mathbb Fpn, 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 kexp(G) related to a conjecture of Kubertin.