Heights, regulators and Schinzel’s determinant inequality
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.