Generalized Pohst inequality and small regulators

TL;DR

This paper improves bounds on the regulators of certain number fields by optimizing polynomial functions over hypercubes, leading to the identification of fields with minimal regulators and expanding known lists.

Contribution

It introduces new upper bounds for regulators of degree 5 to 9 fields with one complex embedding, adapting strategies from the totally real case and overcoming computational challenges.

Findings

Identified four number fields of signature (6,1) with smallest regulator.

Expanded lists of number fields with small regulators in signatures (3,1), (4,1), and (5,1).

Developed effective bounds using polynomial optimization over hypercubes.

Abstract

Current methods for the classification of number fields with small regulator depend mainly on an upper bound for the discriminant, which can be improved by looking for the best possible upper bound of a specific polynomial function over an hypercube. In this paper, we provide new and effective upper bounds for the case of fields with one complex embedding and degree between five and nine: this is done by adapting the strategy we have adopted to study the totally real case, but for this new setting several new computational issues had to be overcome. As a consequence, we detect the four number fields of signature (6,1) with smallest regulator; we also expand current lists of number fields with small regulator in signatures (3,1), (4,1) and (5,1).