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

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