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A program for solving the existential Hilbert's problem 3 The O-Parameter Case 3.1 Blowing up of singularities of vector fields 3.1.1 Polar and directional blow-up. . . 3.1.2 Successive Blow-ups . . . . . . . . 3.1.3 Quasi-homogeneous blow-up and the Newton diagram v vi Contents 3.2 The finiteness result for analytic vector fields on 8 2 • 3.

A cusp type germ of vector fields is a C∞ germ at 0∈ℝ2, whose 2-jet is C∞ conjugate toWe define a submanifold of codimension 5 in the space of germs consisting of germs of cusp type whose 4-jet is C0 equivalent toOur main result can be stated as follows: any local 3-parameter family in (0, 0) ∈ ℝ2 × ℝ3 cutting transversally in (0, 0) is fibre-C0 equivalent to