Counting points of fixed degree and given height over function fields

Abstract

Let k be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let k denote an algebraic closure of k. We count points in projective space ℙn−1(k) with given height and generating an extension of fixed degree d over k. If n>2d+3, then we derive an asymptotic estimate for the number of such points as the height tends to infinity. As an application, we deduce asymptotic estimates concerning certain decomposable forms.