Implicit Function Theorem for Systems of Polynomial Equations with Vanishing Jacobian and Its Application to Flexible Polyhedra and Frameworks

Abstract. 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.