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Singularities of vector fields on $${\mathbb{R}}^ 3$$ determined by their first non-vanishing jet. (English) Zbl 0661.58005
The authors present a result which gives a sufficient condition for a vector field X on $${\mathbb{R}}^ 3$$ to be equivalent at a singularity to the first non-vanishing jet $$j^ kX(p)$$ of X at p. This condition - which only depends on the homogeneous vector field defined by $$j^ kX(p)$$- is stated in terms of the blown-up vector field $$\bar X$$ (which is defined on $$S^ 2\times {\mathbb{R}})$$, and essentially means that there are no saddle-connections for $$\bar X| S^ 2\times \{0\}.$$
The key tool in the proof will be a result of local normal linearization along a codimension 1 submanifold M providing a $$C^ 0$$ conjugacy having a normal derivative along M equal to 1.
Reviewer: P.Bonckaert

##### MSC:
 58A20 Jets in global analysis 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory
##### Keywords:
vector field; singularity; homogeneous vector field
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##### References:
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