Jumps of ternary cyclotomic coefficients

TL;DR

This paper investigates the differences between consecutive coefficients of ternary cyclotomic polynomials, providing criteria for when these differences are exactly one, and establishes a lower bound on the number of nonzero coefficients.

Contribution

It introduces a new criterion for the coefficient difference being one and proves a lower bound on the count of nonzero coefficients in ternary cyclotomic polynomials.

Findings

Criteria for coefficient differences of one

Lower bound of n^{1/3} on nonzero coefficients

Enhanced understanding of coefficient distribution

Abstract

It is known that two consecutive coefficients of a ternary cyclotomic polynomial $\Phi_{pqr}(x)=\sum_k a_{pqr}(k)x^k$ differ by at most one. In this paper we give a criterion on $k$ to satisfy $|a_{pqr}(k)-a_{pqr}(k-1)|=1$. We use this to prove that the number of nonzero coefficients of the $n$th ternary cyclotomic polynomial is greater than $n^{1/3}$.