On the lines passing through two conjugates of a Salem number
Artūras Dubickas, Chris J. Smyth · Mathematical Proceedings of the Cambridge Philosophical Society · 2008
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 ∈ $\overline{\Q}$ 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 α1+α2=α3+α4 in distinct conjugates of a Pisot number. The Pisot number $\al_1\,{=}\,(1+\sqrt{3+2\sqrt{5}})/2$ shows that this equation has such a solution.