## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 51, Number 2 (2007), 597-604.

### Maximal height of divisors of $x\sp n-1$

Carl Pomerance and Nathan C. Ryan

#### Abstract

The size of the coefficients of cyclotomic polynomials is a problem that has been well-studied. This paper investigates the following generalization: suppose $f(x)\in\mathbb{Z}[x]$ is a divisor of $x^n-1$, so that $f(x)$ is the product of the cyclotomic polynomials corresponding to some of the divisors of $n$. We ask about the largest coefficient in absolute value over all such divisors $f(x)$ of $x^n-1$, obtaining a fairly tight estimate for the maximal order of this function.

#### Article information

**Source**

Illinois J. Math., Volume 51, Number 2 (2007), 597-604.

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258138432

**Digital Object Identifier**

doi:10.1215/ijm/1258138432

**Mathematical Reviews number (MathSciNet)**

MR2342677

**Zentralblatt MATH identifier**

1211.11108

**Subjects**

Primary: 12Y05: Computational aspects of field theory and polynomials

Secondary: 11C08: Polynomials [See also 13F20] 11Y70: Values of arithmetic functions; tables 13B25: Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10]

#### Citation

Pomerance, Carl; Ryan, Nathan C. Maximal height of divisors of $x\sp n-1$. Illinois J. Math. 51 (2007), no. 2, 597--604. doi:10.1215/ijm/1258138432. https://projecteuclid.org/euclid.ijm/1258138432