Ideas from algebraic geometry and arithmetic geometry have contributed to recent advances in symbolic algebra, 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 contributed to the discovery of efficient, and efficiently parallelizable, algorithms. In number theory, rational points on abelian varieties defined over finite fieldshave provided a rich new source of groups with arithmetic properties that can be exploited in the design of new algorithms. Functions on algebraic curves provided a new source of error-correcting codes with highly desirable features. This research addresses certain algebraic problems for which the methods of algebraic geometry have contributed significantly in recent years. More specifically, the object is to study certain computational problems involving algebraic curves -- with applications in number theory, cryptography and coding theory. The project will address those methods which have been used to design efficient parallel and sequential algorithms for these problems, and will consider their broader applications in symbolic and algebraic computation.
