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Every Real Algebraic Integer Is a Difference of Two Mahler Measures

Published online by Cambridge University Press:  20 November 2018

Paulius Drungilas  and
Artūras Dubickas
Paulius Drungilas
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, and, Institute of Mathematics and Informatics, Akademijos 4, LT-08663 Vilnius, Lithuania e-mail: pdrungilas@gmail.com
Artūras Dubickas
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania e-mail: arturas.dubickas@maf.vu.lt
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Abstract

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We prove that every real algebraic integer $\alpha$ is expressible by a difference of two Mahler measures of integer polynomials. Moreover, these polynomials can be chosen in such a way that they both have the same degree as that of $\alpha$, say $d$, one of these two polynomials is irreducible and another has an irreducible factor of degree $d$, so that $\alpha =M\left( P \right)-bM\left( Q \right)$ with irreducible polynomials $P,Q\in \mathbb{Z}\left[ X \right]$ of degree $d$ and a positive integer $b$. Finally, if $d\le 3$, then one can take $b=1$.

Keywords

11R0411R0611R0911R3311D09Mahler measuresPisot numbersPell equationabc-conjecture
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 2 , 01 June 2007 , pp. 191 - 195
Copyright
Copyright © Canadian Mathematical Society 2007

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