 
Summary: arXiv:math.MG/0006126v119Jun2000
Implicit Function Theorem for systems of polynomial equations
with vanishing Jacobian and its application to flexible polyhedra
and frameworks
Victor Alexandrov
Sobolev Institute of Mathematics, Novosibirsk90, 630090, Russia. alex@math.nsc.ru
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.
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
1 Introduction
Let F : Rl
ŚRm
Rn
