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

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