In this work the smooth and topological normal forms of the first-order implicit differential equations in the plane near its folded degenerate elementary singular point are found, and thereby, the smooth and topological classification of folded elementary singular points of these equations is completed. It is proved, for example, that the equation is equivalent near its folded singular point of saddlenode type to some equation 4-x 2 + Ax 3 ---y, where A is a real number, in some appropriate smooth coordinate system in the plane with the origin at this point. The number A is the parameter of the normal form.
It is proved in Ann. Math. (2) 115 (1982) 579-595 that, for any germ of holomorphic nondicritic vector field in (C 2 , 0), there exists at least one separatrix (invariant analytic curve containing the origin). In Proc. Amer. Math. Soc. 125 (1997) 2649-2650 a simple criterion was given to find, at each level of the blow-up, a singular point which leads to an analytical invariant curve. In this paper we prove shortly and strictly combinatorially, the existence of a separatrix, and show that for any germ of holomorphic nondicritic vector field in (C 2 , 0), there exists at least one separatrix issuing from each connected component of the exceptional divisor of its nice blow-up with nodal corner points deleted.
We consider small polynomial deformations of integrable systems of the form dF = 0, F ∈ C[x, y] and the first nonzero term Mµ of the displacement functionIt is known that Mµ is an iterated integral of length at most µ. The bound µ depends on the deformation of dF .In this paper we give a universal bound for the length of the iterated integral expressing the first nonzero term Mµ depending only on the geometry of the unperturbed system dF = 0. The result generalizes the result of Gavrilov and Iliev providing a sufficient condition for Mµ to be given by an abelian integral i.e. by an iterated integral of length 1. We conjecture that our bound is optimal.2010 Mathematics Subject Classification. 34C07; 34C05, 34C08.
We consider the class V d n+1 of dicritical germs of holomorphic vector fields in (C 2 , 0) with vanishing n-jet at the origin for n ≥ 1. We prove, under some genericity assumptions, that the formal equivalence of two generic germs implies their analytic equivalence. A similar result is also established for orbital equivalence. Moreover, we give formal, orbitally formal, and orbitally analytic classifications of generic germs in V d n+1 up to a change of coordinates with identity linear part.
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