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No two non-real conjugates of a Pisot number have the same imaginary part


Authors: Artūras Dubickas, Kevin G. Hare and Jonas Jankauskas
Journal: Math. Comp. 86 (2017), 935-950
MSC (2010): Primary 11R06; Secondary 11R09
DOI: https://doi.org/10.1090/mcom/3103
Published electronically: April 13, 2016
MathSciNet review: 3584555
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Abstract: We show that the number $ \alpha =(1+\sqrt {3+2\sqrt {5}})/2$ with minimal polynomial $ x^4-2x^3+x-1$ is the only Pisot number whose four distinct conjugates $ \alpha _1,\alpha _2,\alpha _3,\alpha _4$ satisfy the additive relation $ \alpha _1+\alpha _2=\alpha _3+\alpha _4$. This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the other hand, we prove that similar four term equations $ \alpha _1 = \alpha _2 + \alpha _3+\alpha _4$ or $ \alpha _1 + \alpha _2 + \alpha _3 + \alpha _4 =0$ cannot be solved in conjugates of a Pisot number $ \alpha $. We also show that the roots of the Siegel's polynomial $ x^3-x-1$ are the only solutions to the three term equation $ \alpha _1+\alpha _2+\alpha _3=0$ in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation $ \alpha _1=\alpha _2+\alpha _3$.


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Additional Information

Artūras Dubickas
Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania
Email: arturas.dubickas@mif.vu.lt

Kevin G. Hare
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: kghare@uwaterloo.ca

Jonas Jankauskas
Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania – and – Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: jonas.jankauskas@gmail.com

DOI: https://doi.org/10.1090/mcom/3103
Keywords: Pisot numbers, Mahler's measure, additive relations
Received by editor(s): May 12, 2015
Received by editor(s) in revised form: August 21, 2015
Published electronically: April 13, 2016
Additional Notes: The research of the first and third authors was supported in part by the Research Council of Lithuania Grant MIP-068/2013/LSS-110000-740
The research of the second author was supported by NSERC Grant RGPIN-2014-03154.
Computational support was provided in part by the Canadian Foundation for Innovation, and the Ontario Research Fund.
Article copyright: © Copyright 2016 American Mathematical Society