On the Euler-Poincar\'e characteristic and mixed multiplicities of maximal degrees

TL;DR

This paper explores the properties of mixed multiplicities related to maximal degrees, introducing the Euler-Poincaré characteristic for joint reductions and establishing key positivity and additivity results.

Contribution

It defines the Euler-Poincaré characteristic for joint reductions and characterizes the positivity of mixed multiplicities, advancing understanding of their algebraic properties.

Findings

Defined Euler-Poincaré characteristic for joint reductions

Characterized positivity of mixed multiplicities

Proved additivity and elementary properties of mixed multiplicities

Abstract

This paper defines the Euler-Poincar\'{e} characteristic of joint reductions of ideals which concerns the maximal terms in the Hilbert polynomial; characterizes the positivity of mixed multiplicities in terms of minimal joint reductions; proves the additivity and other elementary properties for mixed multiplicities. The results of the paper together with the results of [17] seem to show a natural and nice picture of mixed multiplicities of maximal degrees.