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.
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
Affiliations des auteurs :
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.
@article{10_4064_aa8253_1_2016,
author = {Shabnam Akhtari and Jeffrey D. Vaaler},
title = {Heights, regulators and {Schinzel{\textquoteright}s} determinant inequality},
journal = {Acta Arithmetica},
pages = {285--298},
year = {2016},
volume = {172},
number = {3},
doi = {10.4064/aa8253-1-2016},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/aa8253-1-2016/}
}
TY - JOUR
AU - Shabnam Akhtari
AU - Jeffrey D. Vaaler
TI - Heights, regulators and Schinzel’s determinant inequality
JO - Acta Arithmetica
PY - 2016
SP - 285
EP - 298
VL - 172
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.4064/aa8253-1-2016/
DO - 10.4064/aa8253-1-2016
LA - en
ID - 10_4064_aa8253_1_2016
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%0 Journal Article
%A Shabnam Akhtari
%A Jeffrey D. Vaaler
%T Heights, regulators and Schinzel’s determinant inequality
%J Acta Arithmetica
%D 2016
%P 285-298
%V 172
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4064/aa8253-1-2016/
%R 10.4064/aa8253-1-2016
%G en
%F 10_4064_aa8253_1_2016
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
