Lower Bounds for Maximum Gap in (Inverse) Cyclotomic Polynomials

TL;DR

This paper establishes several lower bounds for the maximum gap in cyclotomic and inverse cyclotomic polynomials, often exact, and provides conditions for exact formulas and conjectures for these gaps.

Contribution

It introduces new lower bounds for the maximum gap in cyclotomic and inverse cyclotomic polynomials when n is a product of odd primes, including exact formulas and conjectures.

Findings

Lower bounds for g(Φ_n) and g(Ψ_n) when n is a product of odd primes

Exact expressions for g(Ψ_n) under certain conditions

Conjectured exact formulas for g(Φ_n) under specific conditions

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.