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We present a new technique, based on polynomial continuation, for solving sys- tems of n polynomials in N complex variables. The method,allows equations to be introduced one-by-one or in groups, obtaining at each stage a represen- tation of the solution set that can be extended to the next stage until flnally obtaining the solution set for the entire system. At any stage where posi- tive dimensional solution components must be found, they are sliced down to isolated points by the introduction of hyperplanes. By moving,these hyper- planes, one may build up the solution set to an intermediate system in which a union of hyperplanes \regenerates" the intersection of the component,with the variety of the polynomial (or system of polynomials) brought in at the next stage. The theory underlying the approach guarantees that homotopy,paths lead to all isolated solutions, and this capability can be used to generate witness supersets for solution components at any dimension, the flrst step in comput- ing an irreducible decomposition of the solution set of a system of polynomial equations. The method is illustrated on several challenging problems, where it proves advantageous over both the polyhedral homotopy,method,and the diagonal equation-by-equation method, formerly the two leading approaches to solving sparse polynomial systems by numerical continuation. Key words and phrases. Algebraic set, component of solutions, diagonal homotopy, embedding, equation-by-equation solver, generic point, homotopy continuation, irreducible component, numerical irreducible decomposition, nu- merical algebraic geometry, path following, polynomial system, witness point, witness set. Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 (sommese@nd.edu, www.nd.edu/»sommese). This author was supported by the Duncan Chair of the University of Notre Dame, NSF grant DMS-0410047, and NSF grant DMS-0712910

Regeneration homotopies for solving systems of polynomials