A quantitative version of Northcott's theorem on points of bounded height: The function field case

Abstract

Let $K$ be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let $\overline{K}$ denote an algebraic closure of $K$. For given integers $m⩾0$ and $n⩾2$ we count points in projective space $P^{n-1}\left(\overline{K}\right)$ with absolute logarithmic height $m$ and generating an extension of degree $d>2$ over $K$. Specifically, we derive an asymptotic estimate for the number of such points as $m\to \infty$ when $n>d+1$ and orders of growth for the number of such points when $2⩽n⩽d+1$.