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Tougeron's implicit function theorem and Hensel's lemma are well knownrepresentatives concerning 2k-approximation/k-nondegeneracy implying existenceof solutions with identity of order k. This note aims to extend this principleto equations G[z]=0 in Banach spaces, using k-transversality concepts, whichmay geometrically be interpreted as generalized cones spanned by submanifolds,each characterized by a certain expansion rate. The number of manifolds in thecone, as well as their expansion rates, are recursively increased until anappropriate desingularization of the cone is build up with linearizationexpressed by first k+1 derivatives of the singular operator at the base point.Along these lines, a well-defined submersion is constructed in the cone withuniformly bounded inverse when approaching the singularity. The techniques arerestricted to curves, possibly touching by high order the singular locus of G,but ultimately traversing it, in this way defining an isolated singularity ofthe operator family given by the linearization along the curve. The fineresolution of the cone by the manifolds represents an improvement compared tomeasuring the variation of the nonlinear operator exclusively by the overallbehaviour of the determinant. In case of finite dimensions, each half-cone ischaracterized by a constant topological degree that can be used to investigatea solution curve in general position with respect to secondary bifurcation. Thecore of all considerations is given by some characteristic patterns, valid inthe system of undetermined coefficients that allow for detailed analysis of thepower series resulting from plugging the power series of the ansatz into thepower series of the nonlinear operator.

Fine Resolution of k-transversal Cones