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# Every Real Algebraic Integer Is a Difference of Two Mahler Measures

Published:2007-06-01
Printed: Jun 2007
• Paulius Drungilas
• Artūras Dubickas
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## Abstract

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 $\alpha=M(P)-bM(Q)$ with irreducible polynomials $P, Q\in \mathbb Z[X]$ of degree $d$ and a positive integer $b$. Finally, if $d \leqslant 3$, then one can take $b=1$.
 Keywords: Mahler measures, Pisot numbers, Pell equation, $abc$-conjecture
 MSC Classifications: 11R04 - Algebraic numbers; rings of algebraic integers 11R06 - PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11R09 - Polynomials (irreducibility, etc.) 11R33 - Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10] 11D09 - Quadratic and bilinear equations

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