There are no two non-real conjugates of a Pisot number with the same imaginary part
Artūras Dubickas, Kevin G. Hare, Jonas Jankauskas · arXiv (Cornell University) · 2014
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 $\\alpha_1=\\alpha_2+\\alpha_3$.