  We study the action of the nonlinear mapping G[z] between real or complex
Banach spaces in the vicinity of a given curve with respect to possible
linearization, emerging patterns of level sets, as well as existing solutions
of G[z]=0. The results represent local generalizations of the standard implicit
or inverse function theorem and of Newton's Lemma, considering the order of
approximation needed to obtain solutions of G[z]=0.
  The main technical tool is given by Jordan chains with increasing rank, used
to obtain an Ansatz, appropriate for transformation of the nonlinear system to
its linear part. The family of linear mappings is restricted to the case of an
isolated singularity.
  Geometrically, the Jordan chains define a generalized cone around the given
curve, composed of approximate solutions of order 2k with k denoting the
maximal rank of Jordan chains needed to ensure k-surjectivity of the linear
family. Along these lines, the zero set of G[z] in the cone is calculated
immediately, agreeing up to the order of k-1 with the given approximation.
Hence, the results may also be interpreted as a version of Tougeron's implicit
function theorem or Hensel's Lemma in Banach spaces, essentially restricted to
the arc case of a single variable.
  Finally, by considering a left shift of the Jordan chains, the Ansatz can be
modified in a systematic way to obtain a sequence of refined versions of
linearization theorems and Newton Lemmas in Banach spaces.
