Pisot unit generators in number fields

Abstract

Pisot numbers are real algebraic integers bigger than 1, whose other conjugates have all modulus smaller than 1. In this paper we deal with the algorithmic problem of finding the smallest Pisot unit generating a given number field. We first solve this problem in all real fields, then we consider the analogous problem involving the so called complex Pisot numbers and we solve it in all number fields that admit such a generator, in particular all fields without CM, but not only those.

From positional representation of numbers to positional representation of vectors

“…Algebraic, dynamical, topological, geometric and algorithmic properties of the Rényi number systems have been very intensively studied since then, from both theoretical and practical points of view. For example, a suitable choice of a base in an algebraic extension of rational numbers enables to represent all elements of the algebraic field by a finite or eventually periodic string of digits, see [4] and [5]. Further generalisations of numeration systems emerged in the following years.…”

Monogenic Pisot and Anti-Pisot Polynomials

“…The work in this article was motivated by [3,28,29], and, in particular, by the following theorem of Cheng and Zhuang [3]: Theorem 1.1. Let K be a real Galois extension over Q given by its integral basis β 1 , β 2 , .…”