ABSTRACT
Contents 10.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 10.2 Hilbert-Samuel Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 10.3 Joint Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 10.4 Other Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Multiplicities of ideals are useful invariants, which in good rings determine the ideal up to integral closure. Mixed multiplicities are a collection of invariants of several ideals, generalizing multiplicities, and capturing some information on the interactions among ideals. Teissier and Risler [Tei73] were the first to develop mixed multiplicities, in connection with Milnor numbers of isolated hypersurface singularities: the sequences of Milnor numbers obtained by intersecting with general i-planes arise as mixed multiplicities of the ideal generated by the partial derivatives of the defining power series with the ideal corresponding to the point (see Theorem 10.2.5). Rees connected mixed multiplicities to joint reductions (see Theorem 10.3.1).
