Geodesic
Heights, regulators and Schinzel’s determinant inequality
Shabnam Akhtari 1   ; Jeffrey D. Vaaler 2
1 Department of Mathematics University of Oregon Eugene, OR 97403, U.S.A.
2 Department of Mathematics University of Texas Austin, TX 78712, U.S.A.
Acta Arithmetica, Tome 172 (2016) no. 3, pp. 285-298 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/aa8253-1-2016
Keywords: prove inequalities compare size s regulator product heights multiplicatively independent s units upper bound s regulator follows general upper bound determinant real matrix proved schinzel lower bound s regulator follows minkowski theorem successive minima volume formula proved meyer pajor establish similar upper bounds relative regulator extension number fields

Shabnam Akhtari  1   ; Jeffrey D. Vaaler  2

1 Department of Mathematics University of Oregon Eugene, OR 97403, U.S.A.
2 Department of Mathematics University of Texas Austin, TX 78712, U.S.A. 
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Shabnam Akhtari; Jeffrey D. Vaaler. Heights, regulators and Schinzel’s determinant inequality. Acta Arithmetica, Tome 172 (2016) no. 3, pp. 285-298. doi: 10.4064/aa8253-1-2016

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