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# A Generalization of a Theorem of Boyd and Lawton

Published:2012-07-16
Printed: Dec 2013
• Zahraa Issa,
Département de mathématiques et de statistique, Université de Montréal, Montreal, QC H3C 3J7
• Matilde Lalín,
Département de mathématiques et de statistique, Université de Montréal, Montreal, QC H3C 3J7
 Format: LaTeX MathJax PDF

## Abstract

The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral of $\log|P|$ on the unit $n$-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of $\log|P|$ for possibly different $P$'s), multiple Mahler measure (involving products of $\log|P|$ for possibly different $P$'s), and higher Mahler measure (involving $\log^k|P|$).
 Keywords: Mahler measure, polynomial
 MSC Classifications: 11R06 - PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11R09 - Polynomials (irreducibility, etc.)

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