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

TL;DR

This paper extends an inequality for determinants to exterior products of vectors, applies it to $S$-units in number fields, and establishes a sharp bound related to heights, with implications for a conjecture by Rodriguez Villegas.

Contribution

It generalizes an inequality to exterior products, applies it to $S$-units, and proves the bound is sharp up to a constant depending only on the rank.

Findings

Established a new inequality for exterior products of $S$-units.

Derived a bound expressed as a product of heights.

Proved the inequality is sharp up to a constant depending on rank.

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.