Implicit Function Theorem for systems of polynomial equations with   vanishing Jacobian and its application to flexible polyhedra and frameworks
  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. 
Victor Alexandrov (Sobolev Institute of Mathematics)