The Euler-Poincare characteristic and mixed multiplicities

TL;DR

This paper introduces new concepts of mixed multiplicity systems and establishes their relationship with the Euler-Poincare characteristic and Hilbert polynomial differences for graded modules, advancing the theoretical understanding of mixed multiplicities.

Contribution

It defines mixed multiplicity systems and proves their equivalence with the Euler-Poincare characteristic and Hilbert polynomial differences, providing new tools for studying mixed multiplicities.

Findings

Euler-Poincare characteristic equals mixed multiplicity symbol for given systems

Established the relationship between mixed multiplicity systems and Hilbert polynomial differences

Derived new results for mixed multiplicities based on these concepts

Abstract

This paper defines mixed multiplicity systems; the Euler-Poincare characteristic and the mixed multiplicity symbol of $\mathbb{N}^d$-graded modules with respect to a mixed multiplicity system, and proves that the Euler-Poincare characteristic and the mixed multiplicity symbol of any mixed multiplicity system of the type $(k_1,\ldots,k_d)$ and the $(k_1,\ldots,k_d)$-difference of the Hilbert polynomial are the same. As an application, we get results for mixed multiplicities.