arXiv subjects

Mary Ambrosino

Publications and source records attributed to Mary Ambrosino.

2 recordsLinked to original sources

arXiv2020

Maximum gap in cyclotomic polynomials

Cyclotomic polynomials play fundamental roles in number theory, combinatorics, algebra and their applications. Hence their properties have been extensively investigated. In this paper, we study the maximum gap $g$ (maximum of the differences between any two consecutive exponents). In 2012, it was shown that $g\left( Φ_{p_{1}p_{2}}\right) =p_{1} -1$ for primes $p_{2}>p_{1}$. In 2017, based on numerous calculations, the following generalization was conjectured: $g\left( Φ_{mp}\right) =φ(m)$ for square free odd $m$ and prime $p>m$. The main contribution of this paper is a proof of this conjecture.

math.NT↗

arXiv2017

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.

math.NT↗