Abstract
We give general bounds for the reduction numbers of ideals in arbitrary Noetherian rings and multiplicity-dependent bounds for m-primary ideals in a Noetherian local ring (R,m). 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.
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