  In this paper we outline an algorithmic approach to compute Puiseux series
expansions for algebraic surfaces. The series expansions originate at the
intersection of the surface with as many coordinate planes as the dimension of
the surface. Our approach starts with a polyhedral method to compute cones of
normal vectors to the Newton polytopes of the given polynomial system that
defines the surface. If as many vectors in the cone as the dimension of the
surface define an initial form system that has isolated solutions, then those
vectors are potential tropisms for the initial term of the Puiseux series
expansion. Our preliminary methods produce exact representations for solution
sets of the cyclic $n$-roots problem, for $n = m^2$, corresponding to a result
of Backelin.
