\(V\)-sufficiency from the weighted point of view (Q1342815)

Two germs of functions \(f,g : (\mathbb{R}^ n, 0) \to (\mathbb{R}^ p, 0)\) are said to have the same (local) \(v\)-type at 0 \((v\) stands for variety), if the germs at 0 of \(f^{-1} (0)\) and \(g^{-1} (0)\) are homeomorphic. Let \(f: (\mathbb{R}^ n, 0) \to (\mathbb{R}^ p, 0)\) be a \(C^ k\)-function. A very interesting problem is to determine what terms from the Taylor expansion at 0 may be omitted without changing the \(v\)-type determined by \(f\). In this paper we consider the weighted analogue of this problem, and using a new singular Riemannian metric on \(\mathbb{R}^ n\) we give a characterization of \(v\)-sufficiency (Theorem A and Theorem B). Moreover we give a geometric corollary for functions whose components are the sum of at most two weighted homogeneous polynomials (generalizing the case with nondegenerate weighted homogeneous components), and also we give a generalization of a well-known inequality due to Bochnak and Lojasiewicz.