Heights, regulators and Schinzel’s determinant inequality
Volume 172 / 2016
Acta Arithmetica 172 (2016), 285-298 MSC: 11J25, 11R04, 46B04. DOI: 10.4064/aa8253-1-2016 Published online: 15 January 2016
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 $l/k$ of number fields.