Explicit expression for a family of ternary cyclotomic polynomials

TL;DR

This paper derives explicit formulas for a specific family of ternary cyclotomic polynomials and analyzes their nonzero term density, revealing its inverse proportionality to one of the primes.

Contribution

It provides the first explicit expression for these cyclotomic polynomials and a formula for their nonzero term count, advancing understanding of their structure.

Findings

Explicit formulas for $\

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Abstract

In this paper, we give an explicit expression for a certain family of ternary cyclotomic polynomials: specifically $\Phi_{p_{1}p_{2}p_{3}}$, where $p_{1}<p_{2}<p_{3}$ are odd primes such that $p_{2} \equiv1 \mod p_{1}$ and $p_{3} \equiv1 \mod {p_{1}p_{2}}$. As an application of the explicit expressions, we give an exact formula for the number of nonzero terms in the polynomials in the family, which in turn immediately shows that the density (number of non-zeros terms / degree) is roughly inversely proportional to $p_{2}$, when $p_{1}$ is sufficiently large.