The Euler-Poincare characteristic and mixed multiplicities

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 $(k_1,\dots ,k_d)$ and the $(k_1,\dots ,k_d)$-difference of the Hilbert polynomial are the same. As an application, we get results for mixed multiplicities.