Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

On the lines passing through two conjugates of a Salem number

ARTŪRAS DUBICKASa1 and CHRIS SMYTHa2

a1 Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT- 03225, Lithuania. e-mail: arturas.dubickas@maf.vu.lt

a2 School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, James Clerk Maxwell Bld., King's Buildings, Mayfield Road Edinburgh EH 9 3JZ, Scotland. e-mail: c.smyth@ed.ac.uk

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 xs2208 $\overline{\Q}$ 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 α1234 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.

(Received October 13 2006)

(Revised January 05 2007)