Lowerable vector fields for a finitely \(\mathcal{L}\)-determined multigerm (Q1709218)

The authors give a constructive proof to the following result (Theorem 3), which works well in both the real \(\mathcal{C}^{\infty}\) case and the complex analytic case: Let \(f: (\mathbb{K}^n, S)\rightarrow (\mathbb{K}^p,0)\) be a finitely \(\mathcal{L}\)-determined multi-germ. Then, \(T\mathcal{R}_e(f)\cap T\mathcal{L}_e(f)\) is finitely generated as a \(\mathcal{C}_{p,0}\)-module via \(f\). Here \(\mathbb{K}\) denotes \(\mathbb{R}\) or \(\mathbb{C}\) and \(S\) is a finite set consisting of \(r\) distinct points in \(\mathbb{K}^n\).