Upper Bounds for Cyclotomic Numbers
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 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 ab 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: 2019-02-05
Revised: 2019-04-26
Accepted: 2019-05-08
Published online: 2020-02-12
DOI: https://doi.org/10.5802/alco.86
Classification: 11T22,  11C20
Keywords: Finite fields, Cylotomic Fields, Norm Bounds
@article{ALCO_2020__3_1_39_0,
     author = {Do Duc, Tai and Leung, Ka Hin and Schmidt, Bernhard},
     title = {Upper Bounds for Cyclotomic Numbers},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {3},
     number = {1},
     year = {2020},
     pages = {39-53},
     doi = {10.5802/alco.86},
     zbl = {07169932},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2020__3_1_39_0/}
}
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/item/ALCO_2020__3_1_39_0/

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