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Implicit function theorem for systems of polynomial equations with vanishing Jacobian and its application to flexible polyhedra and frameworks. (English) Zbl 0995.58003
The classical implicit function theorem (and its generalizations and modifications) for a function \(f:\mathbb{R}^p \times\mathbb{R}^q \to \mathbb{R}^n\) provide sufficient conditions under which \(f(\dot x,\varphi(x)) \equiv 0\) for some function \(\varphi:\mathbb{R}^p \to\mathbb{R}^q\) in a neighborhood of a zero of \(f\), the most important condition being the maximal rank of the partial derivative at this zero.
In the present paper, the author gives sufficient conditions, in case of a polynomial function \(f\), which apply also if the partial derivative is singular. He also gives necessary conditions for the existence of \(\varphi\), which therefore may serve as sufficient conditions for non-existence.

MSC:
58C15 Implicit function theorems; global Newton methods on manifolds
26B10 Implicit function theorems, Jacobians, transformations with several variables
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
26C10 Real polynomials: location of zeros
68T40 Artificial intelligence for robotics
70B15 Kinematics of mechanisms and robots
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
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