A conjectural improvement for inequalities related to regulators of number fields

TL;DR

This paper explores a conjectural upper bound to improve existing inequalities related to regulators of number fields, aiming to classify fields with small regulators for degrees 5, 7, and 8.

Contribution

It proposes a conjectural polynomial bound that, if true, enhances Remak-Friedman's inequality for better classification of number fields.

Findings

Potential classification of number fields with small regulators in degrees 5, 7, and 8.

Discussion on the validity of the conjectured polynomial bound.

Improved bounds could lead to finite classifications for certain signatures.

Abstract

An inequality proved firstly by Remak and then generalized by Friedman shows that there are only finitely many number fields with a fixed signature and whose regulator is less than a prescribed bound. Using this inequality, Astudillo, Diaz y Diaz, Friedman and Ramirez-Raposo succeeded to detect all fields with small regulators having degree less or equal than 7. In this paper we show that a certain upper bound for a suitable polynomial, if true, can improve Remak-Friedman's inequality and allows a classification for some signatures in degree 8 and better results in degree 5 and 7. The validity of the conjectured upper bound is extensively discussed.