Algorithmic Search for Flexibility Using Resultants of Polynomial Systems

This talk describes the recent convergence of four topics: polynomial systems, flexibility of three dimensional objects, computational chemistry, and computer algebra. We discuss a way to solve systems of polynomial equations with resultants. Using ideas of Bricard, we find a system of polynomial equations that models a configuration of quadralaterals that is equivalent to some three dimensional structures. These structures are of interest in computational chemistry, as they represent molecules. We then describe and demonstrate an algorithm that examines the resultant and determines ways that the structure can be flexible. We review some ideas about flexibility, from Cauchy (1813) to Bricard (1893) to Connelly (1977) to Steffan (1980).