Reduction Numbers of Ideals

We give general bounds for the reduction numbers of ideals in arbitrary Noetherian rings and multiplicity-dependent bounds for -primary ideals in a Noetherian local ring (R, ). In the case of polynomial rings over fields the bound is a non-elementary function with four levels of exponentiation; for primary ideals the bound is linear in the Samuel multiplicity of the ideal. Finally we extend these techniques to generic complete intersections of dimension one.