Jacobian为零的多项式方程组的隐函数定理及其在柔性多面体和框架中的应用
Implicit Function Theorem for systems of polynomial equations with vanishing Jacobian and its application to flexible polyhedra and frameworks
摘要：
研究了由非线性代数方程组确定的局部隐函数在其雅可比矩阵的行列式在所考虑的点处为零的特殊情况下的存在性问题.我们提出了一个系统的充分条件，这意味着存在的一个局部隐函数以及另一个系统的充分条件，保证不存在一个局部隐函数。所得结果被应用于证明新的和经典的结果的灵活性和刚性的多面体和框架。
英文摘要：
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.
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