ALGEBRAIC COMBINATORICS

no. 1
Upper Bounds for Cyclotomic Numbers
Do Duc, Tai1; Leung, Ka Hin2; Schmidt, Bernhard1
1 Division of Mathematical Sciences School of Physical & Mathematical Sciences Nanyang Technological University Singapore 637371 Republic of Singapore
2 Department of Mathematics National University of Singapore Kent Ridge, Singapore 119260 Republic of Singapore
Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 39-53.

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 𝔽 q . A general result of our study is that (a,b)≤3 for all a,b∈ℤ if p>(14) k/ord k (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 𝔽 q 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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.86
Classification: 11T22, 11C20
Keywords: Finite fields, Cylotomic Fields, Norm Bounds
Do Duc, Tai 1; Leung, Ka Hin 2; Schmidt, Bernhard 1

1 Division of Mathematical Sciences School of Physical & Mathematical Sciences Nanyang Technological University Singapore 637371 Republic of Singapore
2 Department of Mathematics National University of Singapore Kent Ridge, Singapore 119260 Republic of Singapore
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Do Duc, Tai; Leung, Ka Hin; Schmidt, Bernhard. Upper Bounds for Cyclotomic Numbers. Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 39-53. doi : 10.5802/alco.86. https://alco.centre-mersenne.org/articles/10.5802/alco.86/

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