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Formal reduction of cuspidal singularities of analytic vector fields. (Réduction formelle des singularités cuspidales de champs de vecteurs analytiques.) (French) Zbl 0985.37014
This paper deals with the formal orbital equivalence of planar vector fields having a singularity of nilpotent type. The main result of the paper asserts that any real or complex one-form $w=d(y^2-x^q)+\Delta(x,y)(2x dy-q y dx),$ where $$\Delta(x,y)$$ does not contain neither constant term nor monomials of the form $$x,x^2,\ldots, x^{[q/2]}$$, is formally equivalent to a one-form of the type $d(y^2-x^q)+\{\Delta_0(h)+x\Delta_1(h)+\cdots+x^{q-2}\Delta_{q-2}(h)\}(2x dy-qy dx),$ where the functions $$\Delta_k$$ are series in the variable $$h=y^2-x^q,$$ without constant term for any $$k=0,1\ldots, [q/2]-1.$$ Here $$[ ]$$ denotes the integer part function. Furthermore this reduction is not unique. In particular, if not all the $$\Delta_k\equiv 0,$$ let $$k_0$$ be such that $$\Delta_{k_0}$$ is the first non zero function. Then it is possible to obtain that $$\Delta_{k_0}(h)=h^{m}+\mu h^{2m},$$ with $$\mu=0$$ if $$k_0\neq q/2$$ or with $$\mu$$ a constant if $$k_0=q/2.$$ In the case $$k_0=q/2$$ it is also possible to eliminate another parameter in the next nonzero $$\Delta_k.$$ As the author also noticed, any nilpotent singularity which has dominant term $$d(y^2-x^q)$$ with the usual weights associated to this Hamiltonian, can be analytically transformed into a one-form $$w$$ of the type considered in the above result.
The final part of the paper is devoted to discuss the convergence of the changes of coordinates used.

##### MSC:
 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 58K50 Normal forms on manifolds 37F75 Dynamical aspects of holomorphic foliations and vector fields
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