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Pisot unit generators in number fields

Tomáš Vávra, Francesco Veneziano

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.

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.

Paper Structure

This paper contains 8 sections, 10 theorems, 22 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Number fields
  4. Salem numbers
  5. Height
  6. Locating Pisot units
  7. The algorithm
  8. Height bound

Key Result

Theorem 3

Algorithm FINDMIN always terminates with output $\alpha$. When $K$ is a real field, and $\phi_k$ is the identity embedding, then $\lvert\alpha\rvert$ is a $U$-number, an algebraic unit, it generates the field $K$ over $\mathbb Q$, and it has the smallest height among elements with this properties.

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 6
  • Proposition 7: See remak §2
  • Theorem 8
  • Definition 9
  • Remark 10
  • ...and 14 more