Upper Bounds for Cyclotomic Numbers

Tai Do Duc,Bernhard Schmidt,Ka Hin Leung

doi:10.5802/alco.86

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 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.

Highlights

Introduction and DefinitionsFirst, we fix some notation and definitions
We study upper bounds for cyclotomic numbers (a, b) of order e√over the finite field Fq
Let g denote a primitive element of the finite field Fq

Summary

Introduction and Definitions

We fix some notation and definitions. By q we denote a power of a prime p. Cyclotomic numbers have been studied for decades by many authors, as they have applications in various areas. Cyclotomic classes Ca were used by Paley [11] in 1993 to construct difference sets This approach was later employed by many other authors. For a positive integer k, let ζk denote a complex primitive kth root of unity. The following result about eigenvalues and eigenvectors of a circulant matrix is well known, see [5], for example. We review some results on vanishing sums of roots of unity which will be needed for our study. Let T be a finite set of complex roots of unity and let cα, α ∈ T , be nonzero rational numbers.

Vanishing Sums of Roots of Unity

Bounds on Norms of Cyclotomic Integers

Equations over Fq and C

Upper Bounds for Cyclotomic Numbers

The Case where k is Prime