First Order Perturbation and Local Stability of Parametrized Systems

A problem frequently encountered in geometric constraint solving and related settings is to ascertain sensitivity of solutions arising from a well constrained input configuration. This is important for tolerancing and motion planning, for example. An example would be determining lines simultaneously tangent to four given spheres (which originates as a line-of-sight problem); how much does a perturbation of the input affect the positioning of these lines. Once translated to an algebraic setting one has a system of polynomial equations with some coefficients parametrized, and wants to determine the solutions and a good approximation of their sensitivity to changes in the parameters. We will compute this sensitivity from first order changes in an appropriate Gröbner basis. We demonstrate the applicability on several examples. We also discuss a more global form of stability, wherein one wants to know about perturbations that might change the character of the solution space e.g. by having fewer than the generic number of solutions.