There are no two non-real conjugates of a Pisot number with the same imaginary part

Abstract

We show that the number $α=(1+\sqrt{3+2\sqrt{5}})/2$ with minimal polynomial $x^4-2x^3+x-1$ is the only Pisot number whose four distinct conjugates $α_1,α_2,α_3,α_4$ satisfy the additive relation $α_1+α_2=α_3+α_4$. 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 $α_1 = α_2 + α_3+α_4$ or $α_1 + α_2 + α_3 + α_4 =0$ cannot be solved in conjugates of a Pisot number $α$. We also show that the roots of the Siegel's polynomial $x^3-x-1$ are the only solutions to the three term equation $α_1+α_2+α_3=0$ in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation $α_1=α_2+α_3$.