A Survey of Some Results for Mixed Multiplicities

TL;DR

This survey unifies various results on mixed multiplicities of ideals, showing that several key sequences are equivalent and extending earlier theorems to a broader context in commutative algebra.

Contribution

It demonstrates that superficial sequences are weak-(FC)-sequences, unifying multiple prior results and extending the scope of mixed multiplicity theorems.

Findings

Superficial sequences are weak-(FC)-sequences.

Unified results for mixed multiplicities of ideals.

Extended earlier theorems to a broader class of sequences.

Abstract

The paper proves that the superficial sequences in [13, 15, 16, 25] are the weak-(FC)-sequences in [18]. This follows that [18, Theorem 3.4] of Viet in 2000 covers the results of Risler and Teissier [13] in 1973; Trung [15, Theorem 3.4] in 2001; Trung and Verma [16, Theorem 1.4] in 2007. The paper unified the results for mixed multiplicities of ideals in [13, 15, 16, 17, 18, 25].