没有两个Pisot数的非实共轭具有相同的虚部
There are no two non-real conjugates of a Pisot number with the same imaginary part
摘要：
我们证明了具有极小多项式$x^4-2x^3+x-1$的数$α=(1+\sqrt{3+2\sqrt{5}})/2$是唯一的四个不同共轭$α_1,α_2,α_3,α_4$满足加性关系$α_1+α_2=α_3+α_4$的Pisot数.这意味着不存在两个具有相同虚部的Pisot数的非实共轭，并且Pisot数的至多两个共轭可以具有相同的实部。另一方面，我们证明了类似的四项方程$α_1 = α_2 + α_3+α_4$或$α_1 + α_2 + α_3 + α_4 =0$不能在Pisot数$α$的共轭中求解。我们还表明，根的西格尔的多项式$x^3-x-1$是唯一的解决方案，三项方程$α_1+α_2+α_3=0$共轭的皮索数。最后证明了不存在共轭满足关系式$α_1=α_2+α_3$的Pisot数.
英文摘要：
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 $α_1=α_2+α_3$.
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