The Recursion Formula for Mixed Multiplicities with respect to Joint Reductions

Abstract: This paper gives the recursion formula for mixed multiplicities of maximal degrees with respect to joint reductions of ideals, which is one of important results in the mixed multiplicity theory. Using this result, we give consequences on the relationship between mixed multiplicities and the Hilbert-Samuel multiplicity under most essential assumptions, that is a substantial progress in the problem of expressing mixed multiplicities into the Hilbert-Samuel multiplicity.