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arXiv · 1912.08512

A conjectural improvement for inequalities related to regulators of number fields

Francesco Battistoni
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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.

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Francesco Battistoni. 2019-12-18. A conjectural improvement for inequalities related to regulators of number fields. https://arxiv.org/abs/1912.08512

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