Sums of heights of algebraic numbers
Gregory P. Dresden · Mathematics of Computation · 2002
For A t ( x ) = f ( x ) − t g ( x ) A_t(x) = f(x) - t\, g(x) , we consider the set { ∑ A t ( α ) = 0 h ( α ) : t ∈ Q ¯ } \{ \sum _{A_t(\alpha ) = 0} h(\alpha ) : t \in \overline {\mathbb {Q}} \} . The polynomials f ( x ) , g ( x ) f(x), g(x) are in Z [ x ] \mathbb {Z}[x] , with only mild restrictions, and h ( α ) h(\alpha ) is the Weil height of α \alpha . We show that this set is dense in [ d , ∞ ) [d, \infty ) for some effectively computable limit point d d .