Cyclotomic System and their Arithmetic

2024/12/17 by Li Zhu, Zhu, Li, Hongfeng Wu +2 · 1 citation
Computer Science · #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2412.12455

Abstract

Let q=p^e be a prime power, ℓ be a prime number different from p, and n be a positive integer divisible by neither p nor ℓ. In this paper we define the ℓ-adic q-cyclotomic system PC(ℓ,q,n) with base module n and the total q-cyclotomic system PC_q, which are projective limits of certain spaces of q-cyclotomic cosets. Comparing to q-cyclotomic cosets modulo a fixed integer, the compatible sequences of q-cyclotomic cosets lying in these systems can be characterized and classified in a natural way. We give a detailedd description of the ℓ-adic q-cyclotomic system in the cases where ℓ is an odd prime and where ℓ=2 respectively. As an application, we represent an algorithm to determine a full set of representatives and the sizes of the cosets with any given parameters.

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• The Minimal Binomial Multiples of Polynomials over Finite Fields

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