No two non-real conjugates of a Pisot number have the same imaginary parts
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in: Mathematics of computation, Jahrgang 86.2017, Nr. 304, 03.2017, S. 935-950.
Publikationen: Beitrag in Fachzeitschrift › Artikel › Forschung › (peer-reviewed)
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TY - JOUR
T1 - No two non-real conjugates of a Pisot number have the same imaginary parts
AU - Dubickas, Artūras
AU - Hare, Kevin
AU - Jankauskas, Jonas
PY - 2017/3
Y1 - 2017/3
N2 - 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.
AB - 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.
KW - Pisot numbers
KW - Linear equations
KW - Additive relations
KW - Mahler's measure
M3 - Article
VL - 86.2017
SP - 935
EP - 950
JO - Mathematics of computation
JF - Mathematics of computation
SN - 1088-6842
IS - 304
ER -