A local to global principle for densities over function fields

Abstract

Let $d$ be a positive integer and $\mathbb H$ be an integrally closed subring of a global function field $F$. The purpose of this paper is to provide a general sieve method to compute densities of subsets of $\mathbb H^d$ defined by local conditions. The main advantage of the method relies on the fact that one can use results from measure theory to extract density results over $\mathbb H^{d}$. Using this method we are able to compute the density of the set of polynomials with coefficients in $\mathbb H$ which give rise to "good" totally ramified extensions of the global function field $F$. As another application, we give a closed expression for the density of rectangular unimodular matrices with coefficients in $\mathbb H$ in terms of the $L$-polynomial of the function field.