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We develop geometry-of-numbers methods to count orbits in prehomogeneousvector spaces having bounded invariants over any global field. As our primaryexample, we apply these techniques to determine, for any base global field $F$of characteristic not 2, the density of discriminants of field extensions ofdegree at most 5 over $F$.

Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces