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Functional degrees and arithmetic applications III: Beyond prime exponent

Part of: Finite commutative rings Representation theory of groups Arithmetic algebraic geometry Abelian groups

Published online by Cambridge University Press:  30 May 2025

PETE L. CLARK  and
UWE SCHAUZ
PETE L. CLARK
Affiliation:
Mathematics Department, University of Georgia Athens, GA 30606, U.S.A. e-mail: plclark@gmail.com
UWE SCHAUZ
Affiliation:
Department of Pure Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou 215123, Jiangsu Province, P. R. China.  e-mail: uwe.schauz@xjtlu.edu.cn
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Abstract

Continuing our work on group-theoretic generalisations of the prime Ax–Katz Theorem, we give a lower bound on the p-adic divisibility of the cardinality of the set of simultaneous zeros $Z(f_1,f_2,\dots,f_r)$ of r maps $f_j\,{:}\,A\rightarrow B_j$ between arbitrary finite commutative groups A and $B_j$ in terms of the invariant factors of $A, B_1,B_2, \cdots,B_r$ and the functional degrees of the maps $f_1,f_2, \dots,f_r$.

MSC classification

Primary: 13M10: Polynomials
Secondary: 20K01: Finite abelian groups 20D60: Arithmetic and combinatorial problems 11G25: Varieties over finite and local fields
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 30
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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