There are no two non-real conjugates of a Pisot number with 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 ${\alpha }_1={\alpha }_2+{\alpha }_3$.

References in corpus (1)

• Lower bounds on the projective heights of algebraic points