Every Real Algebraic Integer Is a Difference of Two Mahler Measures

Drungilas, Paulius ; Dubickas, Artūras

Canadian mathematical bulletin, Tome 50 (2007) no. 2, pp. 191-195

Résumé

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 $b=1$ .

Détail

DOI : 10.4153/CMB-2007-020-0

Mots-clés : 11R04, 11R06, 11R09, 11R33, 11D09, Mahler measures, Pisot numbers, Pell equation, abc-conjecture

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