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

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