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Quaternionic Mahler measure

Weijia Wang, Hao Zhang

Abstract

We introduce the quaternionic Mahler measure for non-commutative polynomials, extending the classical complex Mahler measure. We establish the existence of quaternionic Mahler measure for slice regular polynomials in one and two variables. We study the quaternionic Mahler measure for real and slice regular polynomials, and consider the associated Lehmer problem. Various formulas of quaternionic Mahler measures are proved.

Quaternionic Mahler measure

Abstract

We introduce the quaternionic Mahler measure for non-commutative polynomials, extending the classical complex Mahler measure. We establish the existence of quaternionic Mahler measure for slice regular polynomials in one and two variables. We study the quaternionic Mahler measure for real and slice regular polynomials, and consider the associated Lehmer problem. Various formulas of quaternionic Mahler measures are proved.
Paper Structure (8 sections, 39 theorems, 216 equations)

This paper contains 8 sections, 39 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on quaternions
  3. Quaternion slice regular functions
  4. Jensen formula and Mahler measures of real polynomials
  5. Mahler measures of slice regular polynomials
  6. Examples on non-commutative linear polynomials
  7. Multivariable quaternionic Mahler measures
  8. Quaternion Lehmer problem

Key Result

Theorem 1.1

In the following formulas, let $\alpha\in \mathbb{H}$ and $\phi=\arccos (\mathop{\mathrm{Re}}\nolimits \alpha/|\alpha|)$.

Theorems & Definitions (87)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark
  • Theorem 1.6
  • Definition 3.1
  • Lemma 3.2: Splitting Lemma
  • Lemma 3.3: Identity principle
  • ...and 77 more