Mahler measures in a cubic field

Artūras Dubickas

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 3, page 949-956
  • ISSN: 0011-4642

Abstract

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We prove that every cyclic cubic extension E of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in E . This extends the result of Schinzel who proved the same statement for every real quadratic field E . A corresponding conjecture is made for an arbitrary non-totally complex field E and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure.

How to cite

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Dubickas, Artūras. "Mahler measures in a cubic field." Czechoslovak Mathematical Journal 56.3 (2006): 949-956. <http://eudml.org/doc/31080>.

@article{Dubickas2006,
abstract = {We prove that every cyclic cubic extension $E$ of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in $E$. This extends the result of Schinzel who proved the same statement for every real quadratic field $E$. A corresponding conjecture is made for an arbitrary non-totally complex field $E$ and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure.},
author = {Dubickas, Artūras},
journal = {Czechoslovak Mathematical Journal},
keywords = {Mahler measure; Pisot numbers; cubic extension; Mahler measure; Pisot numbers; cubic extension},
language = {eng},
number = {3},
pages = {949-956},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Mahler measures in a cubic field},
url = {http://eudml.org/doc/31080},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Dubickas, Artūras
TI - Mahler measures in a cubic field
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 3
SP - 949
EP - 956
AB - We prove that every cyclic cubic extension $E$ of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in $E$. This extends the result of Schinzel who proved the same statement for every real quadratic field $E$. A corresponding conjecture is made for an arbitrary non-totally complex field $E$ and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure.
LA - eng
KW - Mahler measure; Pisot numbers; cubic extension; Mahler measure; Pisot numbers; cubic extension
UR - http://eudml.org/doc/31080
ER -

References

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