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On the lines passing through two conjugates of a Salem number

  • ARTŪRAS DUBICKAS (a1) and CHRIS SMYTH (a2)
Abstract
Abstract

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 [d2/2]-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 ∈ of degree d are in general position if and only if α has 2 real conjugates, d-2 complex conjugates, no three distinct conjugates of α lie on a line and any two lines that pass through two distinct conjugates of α are non-parallel, except for d/2-1 lines parallel to the imaginary axis. Our main result asserts that the conjugates of any Salem number are in general position. We also ask two natural questions about conjugates of Pisot numbers which lead to the equation α1234 in distinct conjugates of a Pisot number. The Pisot number shows that this equation has such a solution.

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References
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[1]Mignotte M.. Sur les conjugées des nombres de Pisot. C. R. Acad. Sci. Paris Sér. I. Math 298 (1984), 21.
[2]Serre J.-P.. Topics in Galois Theory. Research Notes in Mathematics 1 (Jones & Bartlett, 1992).
[3]Smyth C. J.. The conjugates of algebraic integers. Amer. Math. Monthl 82 (1975), 86.
[4]Smyth C. J.. Conjugate algebraic numbers on conics. Acta Arith. 40 (1982), 333346.
[5]Salem R.. Algebraic Numbers and Fourier Analysis (D. C. Heath, 1963).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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