The first numerical criterion for integral dependence was proved by Rees in 1961: Let I \subset J  be two m-primary ideals in an equidimensional and universally catenary Noetherian local ring (R, m). Then I and J have the same integral closure if and only if they have the same Hilbert-Samuel multiplicity. This criterion plays an important role in Teissier's work on the equisingularity of families of hypersurfaces with isolated singularities.  For hypersurfaces with non-isolated singularities, one needs a similar numerical criterion for integral dependence of non-m-primary ideals. Since the Hilbert-Samuel multiplicity is no longer defined for non-m-primary ideals, the need arose to use other notions of multiplicities that can be used to check for integral dependence. 

The multiplicity sequence was introduced by Achilles and Manaresi in 1997 and has its origin in the intersection numbers of the Stuckrad-Vogel algorithm in intersection theory. Since then it has been conjectured that two arbitrary ideals I \subset J in an equidimensional and universally catenary Noetherian local ring have the same integral closure if and only if they have the same multiplicity sequence. This conjecture has been recently settled by Polini-Trung-Ulrich-Validashti. This talk will present a history on numerical criterions for integral dependence and outline the proof of the above conjecture.