The Reciprocal Algebraic Integers Having Small House

Abstract

Let α be a reciprocal algebraic integer of degree d. The house of α is the largest modulus of its conjugates. We compute the minimum of the houses of all reciprocal algebraic integers of degree d having the minimal polynomial which is a factor of a Dth degree reciprocal or antireciprocal polynomial with at most eight monomials, say $\text{mr}(d)$, for d at most 180, $D\le 1.5d$ and $D\le 210$. We show that it is not necessary to take into account imprimitive polynomials. The computations suggest several conjectures. We show that d-th power of the house of a sequence of reciprocal primitive polynomials has a limit. We present a property of antireciprocal hexanomials.

KEYWORDS:

• Algebraic integer
• the house of algebraic integer
• maximal modulus
• reciprocal polynomial
• primitive polynomial
• Schinzel-Zassenhaus conjecture
• Mahler measure
• method of least squares
• cyclotomic polynomials

2010 Mathematics Subject Classification:

• 11C08
• 11R06
• 11Y40

Acknowledgments

I am very grateful to an unknown referee for bringing the articles [5, 8] and [10] to my attention, for his kind permission to include his Lemma 2 in this paper, and for many valuable comments.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Funding

Partially supported by Serbian Ministry of Education and Science, Project 174032.