Upper bounds on cyclotomic numbers

TL;DR

This paper establishes upper bounds for cyclotomic numbers over finite fields, providing new theoretical limits based on matrix determinants and binomial coefficients.

Contribution

It introduces novel upper bounds for cyclotomic numbers of order e, improving understanding of their maximum possible values under certain conditions.

Findings

Cyclotomic numbers are at most ⌈k/2⌉ under certain assumptions.

The cyclotomic number (0,0) is at most ⌈k/2⌉ - 1.

Bounds are derived using determinants of matrices with binomial coefficient entries.

Abstract

In this article, we give upper bounds for cyclotomic numbers of order e over a finite field with q elements, where e is a divisor of q-1. In particular, we show that under certain assumptions, cyclotomic numbers are at most $\lceil\frac{k}{2}\rceil$, and the cyclotomic number (0,0) is at most $\lceil\frac{k}{2}\rceil-1$, where k=(q-1)/e. These results are obtained by using a known formula for the determinant of a matrix whose entries are binomial coefficients.