# Positivity of Mixed Multiplicities of Filtrations

## TL;DR

This paper investigates the positivity of mixed multiplicities of filtrations in local rings, establishing conditions for their positivity and providing examples of irrational and zero values, with extensions to modules.

## Contribution

It proves that mixed multiplicities are nonnegative, characterizes when they are positive in analytically irreducible rings, and extends results to modules.

## Key findings

- Mixed multiplicities are always nonnegative real numbers.
- Positivity of mixed multiplicities is equivalent to positivity of individual multiplicities in certain rings.
- Examples show mixed multiplicities can be zero or irrational.

## Abstract

The theory of mixed multiplicities of filtrations by $m$-primary ideals in a ring is introduced in a recent paper by Cutkosky, Sarkar and Srinivasan. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero or even irrational. When $R$ is analytically irreducible, and $\mathcal I(1),\ldots,\mathcal I(r)$ are filtrations of $R$ by $m_R$-primary ideals, we show that all of the mixed multiplicities $e_R(\mathcal I(1)^{[d_1]},\ldots,\mathcal I(r)^{[d_r]};R)$ are positive if and only if the ordinary multiplicities $e_R(\mathcal I(i);R)$ for $1\le i\le r$ are positive. We extend this to modules and prove a simple characterization of when the mixed multiplicities are positive or zero on a finitely generated module.