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

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- by Artūras Dubickas, Kevin G. Hare and Jonas Jankauskas PDF
- Math. Comp.
**86**(2017), 935-950 Request permission

## 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 $\alpha _1=\alpha _2+\alpha _3$.## References

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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
- MR Author ID: 690847
- 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
- MR Author ID: 825362
- ORCID: 0000-0001-9770-7632
- Email: jonas.jankauskas@gmail.com
- 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. - © Copyright 2016 American Mathematical Society
- Journal: Math. Comp.
**86**(2017), 935-950 - MSC (2010): Primary 11R06; Secondary 11R09
- DOI: https://doi.org/10.1090/mcom/3103
- MathSciNet review: 3584555