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Let $d$ be a positive integer and $\mathbb H$ be an integrally closed subringof a global function field $F$. The purpose of this paper is to provide ageneral sieve method to compute densities of subsets of $\mathbb H^d$ definedby local conditions. The main advantage of the method relies on the fact thatone 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 theset of polynomials with coefficients in $\mathbb H$ which give rise to "good"totally ramified extensions of the global function field $F$. As anotherapplication, we give a closed expression for the density of rectangularunimodular matrices with coefficients in $\mathbb H$ in terms of the$L$-polynomial of the function field.

A local to global principle for densities over function fields