# ALGEBRAIC COMBINATORICS

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 $e=\left(q-1\right)/k$. We study upper bounds for cyclotomic numbers $\left(a,b\right)$ of order $e$ over the finite field ${𝔽}_{q}$. A general result of our study is that $\left(a,b\right)\le 3$ for all $a,b\in ℤ$ if $p>{\left(\sqrt{14}\right)}^{k/{\mathrm{ord}}_{k}\left(p\right)}$. More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: $\left(0,0\right),\left(0,a\right),\left(a,0\right),\left(a,a\right)$ and $\left(a,b\right)$, where $a\ne b$ and $a,b\in \left\{1,...,e-1\right\}$. 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.

Revised:
Accepted:
Published online:
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},
pages = {39--53},
publisher = {MathOA foundation},
volume = {3},
number = {1},
year = {2020},
doi = {10.5802/alco.86},
zbl = {07169932},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.86/}
}
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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