Formal reduction of cuspidal singularities of analytic vector fields (Q1818854)

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.