A bound for the exterior product of $S$-units

Abstract

We generalize an inequality for the determinant of a real matrix proved by A. Schinzel, to more general exterior products of vectors in Euclidean space. We apply this inequality to the logarithmic embedding of $S$-units contained in a number field $k$. This leads to a bound for the exterior product of $S$-units expressed as a product of heights. Using a volume formula of P. McMullen we show that our inequality is sharp up to a constant that depends only on the rank of the $S$-unit group but not on the field $k$. Our inequality is related to a conjecture of F. Rodriguez Villegas.