Ideas from algebraic and arithmetic geometry have to recent advances in symbolic algebraic computation, computational number theory, and in the effective geometry of real semi-algebraic sets. In motion-planning and modeling, the reintroduction of classical methods of elimination theory including multivariate resultants, u- resultants and Chow forms have aided in the design of efficient parallelizable algorithms. In number theory, rational points on abelian varieties defined over finite fields have provided a rich new source of groups with arithmetic properties that can be exploited in the design of new algorithms. This study computational problems for which algebraic and algebro- geometric methods have had significant impact. The research further explores the applications of multivariate resultants to problems involving algebraic curves over finite fields and their Jacobians. These methods have been successfully applied in designing factorization-free algorithms which have achieved order-of-magnitude improvements performance in computing arithmetic properties of curves. Methods of elimination theory that have been used to explore the structure of semi-algebraic sets and to construct solutions to systems of polynomial inequalities are also studied. Such methods have been successfully applied to the design of theoretically efficient solutions to very general problems in solid modeling and algorithmic motion-planning. This research focuses these methods on somewhat more constrained problems manufacturing and from molecular modeling involving topological polyhedra with low degree surfaces in low dimensional spaces where algebraic tools, together with the basic paradigms of computational geometry, provide a framework for the design of both theoretically efficient and potentially practical algorithms.
