Vector fields liftable over finitely determined multigerms of corank at most one

TL;DR

This paper introduces an index to measure the behavior of finitely determined multigerms of corank at most one regarding liftable vector fields, and addresses their finite generation, minimal generators, and construction.

Contribution

It provides a new index for analyzing liftable vector fields over multigerms and characterizes conditions for finite generation and generator construction.

Findings

Defined an index $i_1(f)-i_2(f)$ for multigerms

Characterized when the module of liftable vector fields is finitely generated

Developed methods to compute and construct generators

Abstract

In this paper, we propose one index $i_1(f)-i_2(f)$ which measures how well-behaved a given finitely determined multigerm $f: (\mathbb{K}^n,S)\to (\mathbb{K}^p,0)$ $(n\le p)$ of corank at most one is from the viewpoint of liftable vector fields; and we answer the following problems when the index indicates that the given multigerm $f$ is best-behaved. 1) When is the module of vector fields liftable over $f$ finitely generated? 2) How can we characterize the minimal number of generators when the module of vector fields liftable over $f$ is finitely generated? 3) How can we calculate the minimal number of generators when the module of vector fields liftable over $f$ is finitely generated? 4) How can we construct generators when the module of vector fields liftable over $f$ is finitely generated?