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Summary: arXiv:math.MG/0006126
v1
19
Jun
2000
Implicit Function Theorem for systems of polynomial equations
with vanishing Jacobian and its application to
exible polyhedra
and frameworks
Victor Alexandrov
Sobolev Institute of Mathematics, Novosibirsk-90, 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 suÆcient conditions that implies existence of
a local implicit function as well as another system of suÆcient conditions that guarantees absence
of a local implicit function. The results obtained are applied to proving new and classical results on
exibility and rigidity of polyhedra and frameworks.
2000 Mathematics Subject Classi cation: 52C25, 26B10, 26C10, 68T40, 70B15, 41A58
Key words: Flexible polyhedron,
exible framework, in nitesimal bending, approximate solution
to a system of algebraic equations, implicit function
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