Illinois Journal of Mathematics

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

Carl Pomerance and Nathan C. Ryan

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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

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

First available in Project Euclid: 13 November 2009

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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]


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.

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