Bernshtein's theorem provides a generically exact upper bound on the number of isolated solutions a sparse polynomial system can have in C*^n, with C* = C \ { 0 }. When a sparse polynomial system has fewer than this number of isolated solutions some face system must have solutions in C*^n. In this paper we address the process of recovering a certificate of deficiency from a diverging solution path. This certificate takes the form of a face system along with an approximation of a solution to the face system. We apply extrapolation to estimate the cycle number and the face normal. Applications illustrate the practical usefulness of our approach.
keywords : homotopy continuation, polynomial systems, Newton polytopes, Bernshtein bound, cycle number.
AMS(MOS) Classification : 14Q99, 52A39,52B20, 65H10.
Numerical Algorithms 18(1): 91-108, 1998.