Functional degrees and arithmetic applications III: Beyond prime exponent

Uwe Schauz; Pete L. Clark

doi:10.1017/S0305004125000398

Functional degrees and arithmetic applications III: Beyond prime exponent

Uwe Schauz, Pete L. Clark

Department of Pure Mathematics

University of Georgia

Research output: Contribution to journal › Article › peer-review

Abstract

Continuing our work on group-theoretic generalizations 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 $Z(f_1,f_2,\dotsc,f_r)$ 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 $A, B_1,B_2,\dotsc,B_r$ and the \emph{functional degrees} of the maps $f_1,f_2,\dotsc,f_r$.

Original language	English
Pages (from-to)	1-30
Number of pages	30
Journal	Mathematical Proceedings of the Cambridge Philosophical Society
Early online date	30 May 2025
DOIs	https://doi.org/10.1017/S0305004125000398
Publication status	Published - 30 May 2025

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10.1017/S0305004125000398

Cite this

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title = "Functional degrees and arithmetic applications III: Beyond prime exponent",

abstract = "Continuing our work on group-theoretic generalizations 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 $Z(f_1,f_2,\dotsc,f_r)$ 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 $A, B_1,B_2,\dotsc,B_r$ and the \emph{functional degrees} of the maps $f_1,f_2,\dotsc,f_r$.",

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T1 - Functional degrees and arithmetic applications III

T2 - Beyond prime exponent

AU - Schauz, Uwe

AU - Clark, Pete L.

PY - 2025/5/30

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N2 - Continuing our work on group-theoretic generalizations 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 $Z(f_1,f_2,\dotsc,f_r)$ 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 $A, B_1,B_2,\dotsc,B_r$ and the \emph{functional degrees} of the maps $f_1,f_2,\dotsc,f_r$.

AB - Continuing our work on group-theoretic generalizations 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 $Z(f_1,f_2,\dotsc,f_r)$ 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 $A, B_1,B_2,\dotsc,B_r$ and the \emph{functional degrees} of the maps $f_1,f_2,\dotsc,f_r$.

U2 - 10.1017/S0305004125000398

DO - 10.1017/S0305004125000398

M3 - Article

SN - 0305-0041

SP - 1

EP - 30

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

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