A bound for the exterior product of S-units

Akhtari, Shabnam; Vaaler, Jeffrey D.

doi:10.2140/ant.2024.18.1589

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

Shabnam Akhtari and Jeffrey D. Vaaler

Vol. 18 (2024), No. 9, 1589–1617

DOI: 10.2140/ant.2024.18.1589

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.

Keywords

Weil height, exterior products

Mathematical Subject Classification

Primary: 05D99, 11J25, 11R27, 15A75

Milestones

Received: 22 January 2022

Revised: 24 August 2023

Accepted: 12 October 2023

Published: 19 September 2024

Authors

Shabnam Akhtari
Department of Mathematics Pennsylvania State University University Park, PA United States

Jeffrey D. Vaaler
Department of Mathematics University of Texas at Austin Austin, TX United States

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Vol. 18, No. 9, 2024