  We study the existence problem for a local implicit function determined by a
system of nonlinear algebraic equations in the particular case when the
determinant of its Jacobian matrix vanishes at the point under consideration.
We present a system of sufficient conditions that implies existence of a local
implicit function as well as another system of sufficient conditions that
guarantees absence of a local implicit function. The results obtained are
applied to proving new and classical results on flexibility and rigidity of
polyhedra and frameworks.
