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

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?

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