Toward mixed multiplicities and joint reductions

TL;DR

This paper advances the understanding of mixed multiplicities by generalizing key theorems and removing previous assumptions, bringing us closer to expressing mixed multiplicities as Hilbert-Samuel multiplicities of joint reductions.

Contribution

It generalizes existing theorems on mixed multiplicities and joint reductions, removing the need for joint reductions to be systems of parameters.

Findings

Generalizes Rees's theorem on mixed multiplicities.

Removes the hypothesis that joint reductions are systems of parameters.

Brings the problem of expressing mixed multiplicities closer to Hilbert-Samuel multiplicities.

Abstract

In the direction towards the question when mixed multiplicities are equal to the Hilbert-Samuel multiplicity of joint reductions, this paper not only generalizes [28, Theorem 3.1] that covers the Rees's theorem [13, Theorem 2.4], but also removes the hypothesis that joint reductions are systems of parameters in [28, Theorem 3.1]. The results of the paper seem to make the problem of expressing mixed multiplicities into the Hilbert-Samuel multiplicity of joint reductions become closer.