The Euler–Poincaré characteristic of joint reductions and mixed multiplicities

Abstract

This paper defines the Euler–Poincaré 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 [Thanh and Viet, Mixed multiplicities of maximal degrees, J. Korean Math. Soc. 55(3) (2018) 605–622] seem to show a natural and nice picture of mixed multiplicities of maximal degrees.