Generalized splitting theorem for map-germs
Let \({\mathcal E}(n,p)\) denote the algebra of formal (power series) mappings \(({\mathbb{R}}^ n,0)\to({\mathbb{R}}^ p,0)\) (and similarly for \({\mathbb{C}})\), and let \({\mathfrak M}={\mathcal E}(n,1)\). Let \({\mathcal E}_ i(n,p)\subset {\mathcal E}(n,p)\) be the vector space of homogeneous polynomials of degree i. If \(f\in {\mathcal E}(n,p)\), then \(f=f_ k+f_{k+1}+...\) with \(f_ i\in {\mathcal E}_ i(n,p)\). Let \(G_ i\) be a complement linear subspace of \({\mathcal E}_ i(n,p)\cap {\mathfrak M}^ 2\cdot<\partial f_ k/\partial x>\) in \({\mathcal E}_ i(n,p)\). The claim is: There exists a formal diffeomorphism \(\phi:({\mathbb{R}}^ n,0)\to({\mathbb{R}}^ n,0)\) such that \(f\circ \phi =f_ k+g_{k+1}+g_{k+2}+...\) with \(g_ i\in G_ i\). This contains the Morse lemma, the splitting lemma and a parametrized version contains Whitney's fold and cusp normal forms (in the formal category). This is an announcement without proofs.
scientific article; zbMATH DE number 3868268
scientific article