Functional degrees and arithmetic applications II: The Group-Theoretic Prime Ax-Katz Theorem

2023/05/02 by Pete L. Clark, Clark, Pete L., Uwe Schauz +1
Mathematics · #13F20 #20C05 #20K01 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Group Theory (math.GR) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2305.01304

openalex publication_date 2023/05/02 · openalex created_date 2023/05/04 · openalex updated_date 2026/07/28

Abstract

We give a version of Ax-Katz's p-adic congruences and Moreno-Moreno's p-weight refinement that holds over any finite commutative ring of prime characteristic. We deduce this from a purely group-theoretic result that gives a lower bound on the p-adic divisibility of the number of simultaneous zeros of a system of maps f_j: A→ B_j from a fixed ``source'' finite commutative group A of exponent p to varying ``target'' finite commutative p-groups B_j. Our proof combines Wilson's proof of Ax-Katz over \mathbbF_p with the functional calculus of Aichinger-Moosbauer.

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