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Heights, regulators and Schinzel’s determinant inequality

Volume 172 / 2016

Shabnam Akhtari, Jeffrey D. Vaaler Acta Arithmetica 172 (2016), 285-298 MSC: 11J25, 11R04, 46B04. DOI: 10.4064/aa8253-1-2016 Published online: 15 January 2016


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.


  • Shabnam AkhtariDepartment of Mathematics
    University of Oregon
    Eugene, OR 97403, U.S.A.
  • Jeffrey D. VaalerDepartment of Mathematics
    University of Texas
    Austin, TX 78712, U.S.A.

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