Probabilistic Saturations and Alt's Problem in Mechanism Design

Alt’s problem formulated in 1923 is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be formulated as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992.

Since there is still not a proof that all solutions were obtained, we consider upper bounds for Alt’s problem by counting the number of solutions outside of the base locus to a system arising as the general linear combination of polynomials. In particular, we derive effective symbolic and numeric methods for studying such systems using probabilistic saturations that can be employed using both finite fields and floating-point computations.

The methods are probabilistic and we give bounds on the size of finite field required to achieve a desired level of certainty. Both theoretical and computational results and methods will be discussed.

This talk is based on joint work with Jonathan Hauenstein (University of Notre Dame).