Orbital analytic nonequivalence of saddle resonance vector fields in~$(\mathbf C^2,0)$

P. M. Elizarov

Sbornik. Mathematics, Tome 51 (1985) no. 2, pp. 533-547

Résumé

This article examines germs of holomorphic vector fields fo the form $$ z\frac\partial{\partial z}+w(-1+zw+z^2w^2P(z,w))\frac\partial{\partial w} $$ under the assumption that the support of the power series $P(z,w)$ lies either above the bisector of the first quadrant of the integer lattice $\mathbf Z_+^2$, or below it. Necessary conditions (imposed on the coefficients of $P(z,w)$) are formulated for orbital analytic equivalence of vector fields of the type indicated; these are obtained with the help of approximate calculation of the Écalle–Voronin functional moduli for the analytic classification of germs of holomorphic mappings which are monodromy transformations of the vector fields considered. Bibliography: 18 titles.