No two non-real conjugates of a Pisot number have the same imaginary parts
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Authors
Organisational units
External Organisational units
- University of Waterloo
- Vilnius University
Abstract
We show that the number α=(1+3+25√−−−−−−−√)/2 with minimal polynomial x4−2x3+x−1 is the only Pisot number whose four distinct conjugates α1,α2,α3,α4 satisfy the additive relation α1+α2=α3+α4. 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 α1=α2+α3+α4 or α1+α2+α3+α4=0 cannot be solved in conjugates of a Pisot number α. We also show that the roots of the Siegel's polynomial x3−x−1 are the only solutions to the three term equation α1+α2+α3=0 in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation α1=α2+α3.
Details
Original language | English |
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Pages (from-to) | 935-950 |
Number of pages | 16 |
Journal | Mathematics of computation |
Volume | 86.2017 |
Issue number | 304 |
Early online date | 13 Apr 2013 |
Publication status | Published - Mar 2017 |