The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a member of a parameterized family of polynomial systems where the parameter values may be measured with imprecision or arise from prior numerical computations, uncertainty may arise in the structure of the solution set, including the number of isolated solutions, the existence of higher dimensional solution components, and the number of irreducible components along with their multiplicities. The loci where these structures change form a stratification of exceptional algebraic sets in the space of parameters. We describe methodologies for making the interpretation of numerical results more robust by searching for nearby parameter values on an exceptional set. We demonstrate these techniques on several illustrative examples and then treat several more substantial problems arising from the kinematics of mechanisms and robots.

翻译：数值代数几何领域包含用于数值求解多项式方程组的算法。当方程组是精确的（例如具有有理系数）时，解集是明确定义的。然而，对于参数化多项式系统族中的某个成员，若参数值测量不精确或源于先前的数值计算，则解集的结构可能出现不确定性，包括孤立解的数量、更高维解分量的存在性、不可约分量的数量及其重数。这些结构发生变化的轨迹在参数空间中构成了特殊代数集的分层。我们提出了一种方法，通过搜索特殊集上的邻近参数值，使数值结果的解释更具鲁棒性。我们通过若干说明性示例展示了这些技术，然后处理了几个来自机构与机器人运动学的更具实质性的问题。