Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Fiber Products, exceptional varieties, and mechanisms by Charles WAMPLER, General Motors Research and Development Center Abstract: We consider parameterized algebraic sets, that is, the solution sets of systems of polynomial equations with parameters. An example is the system of loop closure equations for a robot or mechanism, in which the parameters describe the shape of the links, such as link lengths, and the variables describe the motion of the mechanism. It can happen that for certain exceptional subsets of the parameter space, the inverse fibers increase in dimension, and this can be of considerable interest in applications. In the case of mechanisms, these sets are families of mechanisms having exceptional mobility; for example, an exceptional mechanism may have a one dimensional motion curve even though its generic neighbors in the parameter space are immobile structures. We will discuss how fiber products can be used to reveal exceptional sets as irreducible components of an augmented polynomial system derived from the original system in an elementary way. This allows these sets to be systematically discovered using techniques from numerical algebraic geometry. A numerical experiment to find five-bar mechanisms with exceptional mobility will be discussed.