Abstract:
Let $\mathcal{V}_n$ be the class of germs of holomorphic non-dicritic vector fields in $(\mathbb{C}^2,0)$ with vanishing $(n-1)$-jet at the origin, $n\geq2$, and non-vanishing $n$-jet. In the present work the formal normal form (under the strict orbital classification) of generic germs in a subclass $\mathcal{V}_n^o$ of $\mathcal{V}_n$ is given. Any such normal form is given as the sum of three terms: a “principal” generic homogeneous term, $\mathbf{v}_o\in\mathcal{V}_n$, a “hamiltonian” term, $\mathbf{v}_c $ (given by a hamiltonian polynomial vector field) and a “radial” term.
For any generic germ $\mathbf{v}\in\mathcal{V}_n^o$ we define the triplet $i_\mathbf{v}= (\mathbf{v}_o, \mathbf{v}_c,[G_{\mathbf{v}}])$, where $\mathbf{v}_o$ and $\mathbf{v}_c$ denote the principal and hamiltonian terms of its corresponding formal normal form, and $[G_{\mathbf{v}}]$ denotes the class of strict analytic conjugacy of its projective (hidden or vanishing) monodromy group. We prove that the terms appearing in $i_{\mathbf{v}}$ are Thom's invariants of the strict analytical orbital classification of generic germs in $\mathcal{V}_n^o$: two generic germs $\mathbf{v}$ and $\tilde{\mathbf{v}}$ in $\mathcal{V}_n^o$ are strictly orbitally analytically equivalent if and only if $i_{\mathbf{v}}= i_{\tilde{\mathbf{v}}}$. Moreover, any triplet satisfying some natural conditions of concordance can be realized as invariant of a generic germ of $\mathcal{V}_n^o$.
Key words and phrases:Non-dicritic foliations, non-dicritic vector fields, formal normal forms, analytic invariants, monodromy group.
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