  Tougeron's implicit function theorem and Hensel's lemma are well known
representatives concerning 2k-approximation/k-nondegeneracy implying existence
of solutions with identity of order k. This note aims to extend this principle
to equations G[z]=0 in Banach spaces, using k-transversality concepts, which
may geometrically be interpreted as generalized cones spanned by submanifolds,
each characterized by a certain expansion rate. The number of manifolds in the
cone, as well as their expansion rates, are recursively increased until an
appropriate desingularization of the cone is build up with linearization
expressed 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 with
uniformly bounded inverse when approaching the singularity. The techniques are
restricted to curves, possibly touching by high order the singular locus of G,
but ultimately traversing it, in this way defining an isolated singularity of
the operator family given by the linearization along the curve. The fine
resolution of the cone by the manifolds represents an improvement compared to
measuring the variation of the nonlinear operator exclusively by the overall
behaviour of the determinant. In case of finite dimensions, each half-cone is
characterized by a constant topological degree that can be used to investigate
a solution curve in general position with respect to secondary bifurcation. The
core of all considerations is given by some characteristic patterns, valid in
the system of undetermined coefficients that allow for detailed analysis of the
power series resulting from plugging the power series of the ansatz into the
power series of the nonlinear operator.
