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On the absolute Mahler measure of polynomials having all zeros in a sector. III


Authors: V. Flammang and G. Rhin
Journal: Math. Comp. 84 (2015), 2927-2938
MSC (2010): Primary 11R04, 12D10
DOI: https://doi.org/10.1090/mcom/2959
Published electronically: April 22, 2015
MathSciNet review: 3378854
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Abstract: Let $ \alpha $ be an algebraic integer of degree $ d$, not 0 or a root of unity, all of whose conjugates $ \alpha _i$ lie in a sector $ \vert \arg z \vert \leq \theta $. In 1995, G. Rhin and C. Smyth computed the greatest lower bound $ c(\theta )$ of the absolute Mahler measure ( $ \prod _{i=1}^d \max (1, \vert \alpha _i \vert))^{1/d}$ of $ \alpha $, for $ \theta $ belonging to nine subintervals of $ [0, 2 \pi /3]$. More recently, in 2004, G. Rhin and Q. Wu improved the result to thirteen subintervals of $ [0, \pi ]$ and extended some existing subintervals. In this paper, for the first time we find a complete subinterval where $ c(\theta )$ is known exactly, as well as a fourteenth subinterval. Moreover, we slightly extend further all the existing subintervals.


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Additional Information

V. Flammang
Affiliation: UMR CNRS 7502. IECL, Université de Lorraine, site de Metz, Département de Mathématiques, UFR MIM, Ile du Saulcy, CS 50128, 57045 METZ cedex 01, France
Email: valerie.flammang@univ-lorraine.fr

G. Rhin
Affiliation: UMR CNRS 7502. IECL, Université de Lorraine, site de Metz, Département de Mathématiques, UFR MIM, Ile du Saulcy, CS 50128, 57045 METZ cedex 01, France
Email: georges.rhin@univ-lorraine.fr

DOI: https://doi.org/10.1090/mcom/2959
Received by editor(s): January 2, 2014
Received by editor(s) in revised form: March 24, 2014
Published electronically: April 22, 2015
Article copyright: © Copyright 2015 American Mathematical Society