The House of a Reciprocal Algebraic Integer

TL;DR

This paper investigates the minimal house of reciprocal algebraic integers of degree up to 34, providing computational evidence and conjectures related to the Schinzel-Zassenhaus conjecture, and introduces methods involving the powers of the house.

Contribution

It computes minimal houses for reciprocal algebraic integers up to degree 34 and proposes conjectures supported by computational results, advancing understanding of algebraic integer houses.

Findings

Computed minimal houses for degrees up to 34

Formulated conjectures related to the Schinzel-Zassenhaus conjecture

Demonstrated the utility of powers of the house in analysis

Abstract

Let $\alpha$ be an algebraic integer of degree $d$, which is reciprocal. The house of $\alpha$ is the largest modulus of its conjugates. We compute the minimum of the houses of all reciprocal algebraic integers of degree $d$ which are not roots of unity, say $\mathrm{mr}(d)$, for $d$ at most 34. We prove several lemmata and use them to avoid unnecessary calculations. The computations suggest several conjectures. The direct consequence of the last one is the conjecture of Schinzel and Zassenhaus. We demonstrate the utility of $d$-th power of the house of $\alpha$.