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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Integer transfinite diameter and polynomials with small Mahler measure


Authors: Valérie Flammang, Georges Rhin and Jean-Marc Sac-Épée
Journal: Math. Comp. 75 (2006), 1527-1540
MSC (2000): Primary 11Y40, 11R06
Published electronically: March 28, 2006
MathSciNet review: 2219043
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Abstract: In this work, we show how suitable generalizations of the integer transfinite diameter of some compact sets in $ \mathbb{C}$ give very good bounds for coefficients of polynomials with small Mahler measure. By this way, we give the list of all monic irreducible primitive polynomials of $ \mathbb{Z}[X]$ of degree at most $ 36$ with Mahler measure less than $ 1.\,324...$ and of degree $ 38$ and $ 40$ with Mahler measure less than $ 1.\,31$.


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Additional Information

Valérie Flammang
Affiliation: UMR CNRS 7122, Département de Mathématiques, UFR MIM, Université de Metz, Ile du Saulcy, 57045 Metz Cedex 01, France
Email: flammang@poncelet.univ-metz.fr

Georges Rhin
Affiliation: UMR CNRS 7122, Département de Mathématiques, UFR MIM, Université de Metz, Ile du Saulcy, 57045 Metz Cedex 01, France
Email: rhin@poncelet.univ-metz.fr

Jean-Marc Sac-Épée
Affiliation: UMR CNRS 7122, Département de Mathématiques, UFR MIM, Université de Metz, Ile du Saulcy, 57045 Metz Cedex 01, France
Email: jmse@poncelet.univ-metz.fr

DOI: http://dx.doi.org/10.1090/S0025-5718-06-01791-1
PII: S 0025-5718(06)01791-1
Keywords: Mahler measure, integer transfinite diameter, explicit auxiliary function
Received by editor(s): November 24, 2004
Received by editor(s) in revised form: February 8, 2005
Published electronically: March 28, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.