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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
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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