O ct 2 01 4 THERE ARE NO TWO NON - REAL CONJUGATES OF A PISOT NUMBER WITH THE SAME IMAGINARY PART
نویسنده
چکیده
We show that the number α = (1+ √ 3 + 2 √ 5)/2 with minimal polynomial x4−2x3+x−1 is the only Pisot number whose four distinct conjugates α1, α2, α3, α4 satisfy the additive relation α1 + α2 = α3 + α4. 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 α1 = α2 + α3 + α4 or α1 + α2 + α3 + α4 = 0 cannot be solved in conjugates of a Pisot number α. We also show that the roots of the Siegel’s polynomial x − x − 1 are the only solutions to the three term equation α1 + α2 + α3 = 0 in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation α1 = α2 + α3.
منابع مشابه
No two non-real conjugates of a Pisot number have the same imaginary part
We show that the number α = (1 + √ 3 + 2 √ 5)/2 with minimal polynomial x4 − 2x3 + x − 1 is the only Pisot number whose four distinct conjugates α1, α2, α3, α4 satisfy the additive relation α1+α2 = α3+α4. 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 ...
متن کاملOn the lines passing through two conjugates of a Salem number
We show that the number of distinct non-parallel lines passing through two conjugates of an algebraic number α of degree d ≥ 3 is at most [d/2] − d + 2, its conjugates being in general position if this number is attained. If, for instance, d ≥ 4 is even, then the conjugates of α ∈ Q of degree d are in general position if and only if α has 2 real conjugates, d− 2 complex conjugates, no three dis...
متن کاملPisot and Salem Numbers in Intervals of the Real Line
Based on the work of Dufresnoy and Pisot, we develop an algorithm for determining all the Pisot numbers in an interval of the real line, provided this number is finite. We apply the algorithm to the problem of determining small Salem numbers by Salem's construction, and to the proof that certain Pisot sequences satisfy no linear recurrence relation. Introduction. A real algebraic integer 0 > 1 ...
متن کاملWave Propagation Approach to Fluid Filled Submerged Visco-Elastic Finite Cylindrical Shells
Multi-layer orthotropic finite cylindrical shells with a viscoelastic core in contact with fluids are gaining increasing importance in engineering. Vibrational control of these structures is essential at higher modes. In this study, an extended version of the wave propagation approach using first-order shear deformation theory of shell motion is employed to examine the free vibration of damped ...
متن کاملOptical properties of silicon nano layers by using Kramers- Kronig method
Silicon thin layers are deposited on glass substrates with the thickness of 103 nm, 147 nm and 197 nm. The layers are produced with electron gun evaporation method under ultra-high vacuum condition. The optical Reectance and the Transmittance of produced layers were measured by using spectrophotometer. The optical functions such as, real and imaginary part of refractive index, real and imaginar...
متن کامل