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- W4297788130 abstract "We show that the number $alpha=(1+sqrt{3+2sqrt{5}})/2$ 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 $alpha_1=alpha_2+alpha_3$." @default.
- W4297788130 created "2022-10-01" @default.
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- W4297788130 date "2014-10-06" @default.
- W4297788130 modified "2023-10-18" @default.
- W4297788130 title "There are no two non-real conjugates of a Pisot number with the same imaginary part" @default.
- W4297788130 doi "https://doi.org/10.48550/arxiv.1410.1600" @default.
- W4297788130 hasPublicationYear "2014" @default.
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