Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution of a generic system that is used to start up the deformations. Software and applications are discussed.
keywords : polynomial system, numerical algebraic geometry, homotopy, continuation, deformation, path following, dense, sparse, determinantal, B\'ezout bound, Newton polytope, mixed volume, root count, enumerative geometry, numerical Schubert calculus.
AMS Subject Classification : 14N10, 14M15, 52A39, 52B20, 52B55, 65H10, 68Q40.