Heights, Regulators and Schinzel's determinant inequality

TL;DR

This paper establishes new inequalities relating S-regulators and heights of S-units, using determinant bounds and Minkowski's theorem, with implications for number field extensions.

Contribution

It introduces novel bounds for S-regulators based on Schinzel's determinant inequality and Minkowski's theorem, extending to relative regulators of number field extensions.

Findings

Upper bounds for S-regulators derived from determinant inequalities.

Lower bounds for S-regulators using Minkowski's theorem.

Similar bounds established for relative regulators of number fields.

Abstract

We prove inequalities that compare the size of an S-regulator with a product of heights of multiplicatively independent S-units. Our upper bound for the S-regulator follows from a general upper bound for the determinant of a real matrix proved by Schinzel. The lower bound for the S-regulator follows from Minkowski's theorem on successive minima and a volume formula proved by Meyer and Pajor. We establish similar upper bounds for the relative regulator of an extension of number fields.