  A polyhedral method to solve a system of polynomial equations exploits its
sparse structure via the Newton polytopes of the polynomials. We propose a
hybrid symbolic-numeric method to compute a Puiseux series expansion for every
space curve that is a solution of a polynomial system. The focus of this paper
concerns the difficult case when the leading powers of the Puiseux series of
the space curve are contained in the relative interior of a higher dimensional
cone of the tropical prevariety. We show that this difficult case does not
occur for polynomials with generic coefficients. To resolve this case, we
propose to apply polyhedral end games to recover tropisms hidden in the
tropical prevariety.
