On the effective reduction of an ideal

TL;DR

This paper establishes a new condition for the reduction of -primary ideals in Noetherian local rings, extending known results to a broader class of rings by analyzing the sum of degrees of prime divisors.

Contribution

It provides a generalized criterion for ideal reduction based on the degrees of prime divisors in the fiber cone, applicable to any Noetherian local ring.

Findings

Condition for ideal reduction in terms of prime divisor degrees

Extension of known results beyond rings with infinite residue fields

Applicable to all Noetherian local rings

Abstract

It is well known that in the Noetherian local ring with infinite residue field the reduction of $\mm$-primary ideal may be given in the form of a sufficiently general linear combination of its generators. In the paper we give a condition for the existence of such reduction in terms of the sum of degrees of the ideal fiber cone prime divisors in the case of any Noetherian local ring.