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To answer a question of Professor Georges Rhin

V. Flammang

Abstract

Professor Georges Rhin considers a nonzero algebraic integer $\a$ with conjugates $\a_1=\a, \ldots, \a_d$ and asks what can be said about $\d \sum_{ | \a_i | >1} | \a_i |$, that we denote ${\rm{R}}(\a)$. If $\a$ is supposed to be a totally positive algebraic integer, we can establish an analog to the famous Schur-Siegel-Smyth trace problem for this measure. After that, we compute the greatest lower bound $c(θ)$ of the quantities ${\rm{R(\a)}}/d$, for $\a$ belonging to nine subintervals of $]0, 90 [$. The third three subintervals are complete and consecutive. All our results are obtained by using the method of explicit auxiliary functions. The polynomials involved in these functions are found by our recursive algorithm.

To answer a question of Professor Georges Rhin

Abstract

Professor Georges Rhin considers a nonzero algebraic integer with conjugates and asks what can be said about , that we denote . If is supposed to be a totally positive algebraic integer, we can establish an analog to the famous Schur-Siegel-Smyth trace problem for this measure. After that, we compute the greatest lower bound of the quantities , for belonging to nine subintervals of . The third three subintervals are complete and consecutive. All our results are obtained by using the method of explicit auxiliary functions. The polynomials involved in these functions are found by our recursive algorithm.
Paper Structure (8 sections, 3 theorems, 35 equations)

This paper contains 8 sections, 3 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1
  3. Proof of Theorem 2
  4. The principle of auxiliary functions
  5. Link between auxiliary functions and generalized integer transfinite diameter
  6. Construction of an auxiliary function
  7. Optimization of the $c_j$
  8. Proof of Theorem 3

Key Result

Theorem 1

where $\beta_n^2$ is a totally positive algebraic integer of degree $2^n$, defined by C.J. Smyth S1 as follows:

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3