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We develop a method for determining the density of squarefree values taken bycertain multivariate integer polynomials that are invariants for the action ofan algebraic group on a vector space. The method is shown to apply to thediscriminant polynomials of various prehomogeneous and coregularrepresentations where generic stabilizers are finite. This has applications toa number of arithmetic distribution questions, e.g., to the density of smalldegree number fields having squarefree discriminant, and the density of certainunramified nonabelian extensions of quadratic fields. In separate works, themethod forms an important ingredient in establishing lower bounds on theaverage orders of Selmer groups of elliptic curves.

The geometric sieve and the density of squarefree values of invariant polynomials