On the lines passing through two conjugates of a Salem number

Abstract

We show that the number of distinct non-parallel lines passing through two conjugates of an algebraic number alpha of degree d >= 3 is at most [d(2)/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 01 G U of degree d are in general position if and only if a has 2 real conjugates, d - 2 complex conjugates, no three distinct conjugates of a lie on a line and any two lines that pass through two distinct conjugates of a 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 alpha(1) + alpha(2) = alpha(3) + alpha(4) in distinct conjugates of a Pisot number. The Pisot number alpha(1) = (1 + root 3+2 root 5-)/2 shows that this equation has such a solution.