The Recursion Formula for Mixed Multiplicities with respect to Joint Reductions

TL;DR

This paper establishes a recursion formula for mixed multiplicities related to joint reductions of ideals, advancing the understanding of their relationship with Hilbert-Samuel multiplicities in algebraic geometry.

Contribution

It introduces a recursion formula for mixed multiplicities with respect to joint reductions, a significant development in mixed multiplicity theory.

Findings

Derived a recursion formula for mixed multiplicities.

Connected mixed multiplicities with Hilbert-Samuel multiplicity.

Provided new insights into expressing mixed multiplicities in algebraic geometry.

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.