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Every Real Algebraic Integer Is a Difference of Two Mahler Measures |
Canad. Math. Bull. 50(2007), 191-195
Published:2007-06-01
Printed: Jun 2007
Printed: Jun 2007
- Paulius Drungilas
- Artūras Dubickas
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 |