An inequality in mixed multiplicities

TL;DR

This paper generalizes a Minkowski type inequality for mixed multiplicities of filtrations in Noetherian local rings, extending previous results and deriving new inequalities that unify and strengthen existing theories.

Contribution

It introduces a generalized Minkowski inequality for mixed multiplicities of filtrations, broadening the scope of prior results in the field.

Findings

Established a new Minkowski type inequality for mixed multiplicities.

Recovered a known result as a special case of the new inequality.

Extended the theory to non-Noetherian filtrations in local rings.

Abstract

The theory of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, has been developed by Cutkosky, Sarkar and Srinivasan. The objective of this article is to generalise a Minkowski type inequality given in their paper. We also recover a result of Cutkosky, Srinivasan and Verma as a simple consequence of our inequality