Lower Bounds for Maximum Gap in (Inverse) Cyclotomic Polynomials

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 $Φ_{n}$ and $Ψ_{n}$ denote the $n$-th cyclotomic and $n$-th inverse cyclotomic polynomial, respectively. In this paper, we give several lower bounds for $g(Φ_{n})$ and $g(Ψ_{n})$, where $n$ is the product of odd primes. We observe that they are very often exact. We also give an exact expression for $g(Ψ_{n})$ under a certain condition. Finally we conjecture an exact expression for $g(Φ_{n})$ under a certain condition.