Maximum Gap in (Inverse) Cyclotomic Polynomial

Abstract

Let $g(f)$ denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial $f$. Let $Φ_n$ denote the $n$-th cyclotomic polynomial and let $Ψ_n$ denote the $n$-th inverse cyclotomic polynomial. In this note, we study $g(Φ_n)$ and $g(Ψ_n)$ where $n$ is a product of odd primes, say $p_1 < p_2 < p_3$, etc. It is trivial to determine $g(Φ_{p_1})$, $g(Ψ_{p_1})$ and $g(Ψ_{p_1p_2})$. Hence the simplest non-trivial cases are $g(Φ_{p_1p_2})$ and $g(Ψ_{p_1p_2p_3})$. We provide an exact expression for $g(Φ_{p_1p_2}).$ We also provide an exact expression for $g(Ψ_{p_1p_2p_3})$ under a mild condition. The condition is almost always satisfied (only finite exceptions for each $p_1$). We also provide a lower bound and an upper bound for $g(Ψ_{p_1p_2p_3})$.