Lower Bounds for Maximum Gap in (Inverse) Cyclotomic Polynomials

Abstract

The maximum gap $g(f)$ of a polynomial $f$ is the maximum of the differences (gaps) between two consecutive exponents that appear in $f$. Let ${\Phi }_n$ and ${\Psi }_n$ denote the $n$-th cyclotomic and $n$-th inverse cyclotomic polynomial, respectively. In this paper, we give several lower bounds for $g({\Phi }_n)$ and $g({\Psi }_n)$, where $n$ is the product of odd primes. We observe that they are very often exact. We also give an exact expression for $g({\Psi }_n)$ under a certain condition. Finally we conjecture an exact expression for $g({\Phi }_n)$ under a certain condition.