An algorithm for computing the Hilbert--Samuel multiplicities and reductions of zero-dimensional ideals of Cohen--Macaulay local rings

TL;DR

This paper introduces an algorithm to compute minimal reductions and Hilbert--Samuel multiplicities of zero-dimensional ideals in Cohen--Macaulay local rings, aiding in understanding their algebraic structure.

Contribution

The paper presents a novel algorithm for computing reductions and multiplicities of zero-dimensional ideals in Cohen--Macaulay local rings.

Findings

Successfully computes minimal reductions of $rak{m}$-primary ideals.

Determines Hilbert--Samuel multiplicities efficiently.

Solves the membership problem for integral closures of ideals.

Abstract

In this paper, we present an algorithm for computing the minimal reductions of $\mathfrak{m}$-primary ideals of Cohen--Macaulay local rings. Using this algorithm, we are able to compute the Hilbert--Samuel multiplicities and solve the membership problem for the integral closure of $\mathfrak{m}$-primary ideals.