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Title: Upper bounds for cyclotomic numbers
Authors: Duc, Tai Do
Leung, Ka Hin
Schmidt, Bernhard
Keywords: Science::Mathematics
Issue Date: 2020
Source: Duc, T. D., Leung, K. H., & Schmidt, B. (2020). Upper bounds for cyclotomic numbers. Algebraic Combinatorics, 3(1), 39-53. doi:10.5802/alco.86
Project: R-146-000-276-114
RG27/18 (S)
Journal: Algebraic Combinatorics
Abstract: Let q be a power of a prime p, let k be a nontrivial divisor of q−1 and write e=(q−1)/k. We study upper bounds for cyclotomic numbers (a,b) of order e over the finite field Fq. A general result of our study is that (a,b)≤3 for all a,b∈Z if p>(14−−√)k/ordk(p). More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: (0,0),(0,a),(a,0),(a,a) and (a,b), where a≠b and a,b∈{1,...,e−1}. The main idea we use is to transform equations over Fq into equations over the field of complex numbers on which we have more information. A major tool for the improvements we obtain over known results is new upper bounds on the norm of cyclotomic integers.
ISSN: 2589-5486
DOI: 10.5802/alco.86
Rights: © 2020 The journal and the authors. Some rights reserved. This article is licensed under the CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
Fulltext Permission: open
Fulltext Availability: With Fulltext
Appears in Collections:SPMS Journal Articles

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