Polar Multiplicities and Integral Dependence

Abstract We provide new criteria for the integrality and birationality of an extension of graded algebras in terms of the general notion of polar multiplicities of Kleiman and Thorup. As an application, we obtain a new criterion for when a module is a reduction of another in terms of certain mixed Buchsbaum–Rim multiplicities. Furthermore, we prove several technical results regarding polar multiplicities.