Monatshefte für Mathematik

, Volume 132, Issue 4, pp 269–288 | Cite as

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

  • Victor Alexandrov


 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.

2000 Mathematics Subject Classification: 52C25 26B10 26C10 68T40 70B15 41A58 
Key words: Flexible polyhedron flexible framework infinitesimal bending approximate solution to a system of algebraic equations implicit function 


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Copyright information

© Springer-Verlag Wien 2001

Authors and Affiliations

  • Victor Alexandrov
    • 1
  1. 1.Sobolev Institute of Mathematics, Novosibirsk, RussiaRU

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