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

We show that the number α = (1 + √ 3 + 2 √ 5)/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.