## An elementary proof for the upper bound of Remak’s inequality

par - publié le

Vendredi 04 décembre 2020
Francesco Battistoni
(Université de Bourgogne Franche-Comté)

The goal of this work is to study a specific multivariate polynomial and its maximum over the unitary cube : this is required in order to obtain the correct upper bound for an estimate which involves the discriminant and the regulator of a totally real number field. This study was started by Remak and further results were obtained by Pohst and Bertin.
In this seminar we illustrate an elementary procedure which allows to detect the correct value of the maximum : the key idea is to give a graphical representation of the problem for which the research of the maximum is equivalent to prove that every such graphical representation can be transformed (via suitable rules which depend only on simple analytical inequalities) in a representation with a very specific form.