  This paper investigates the cost of solving systems of sparse polynomial
equations by homotopy continuation. First, a space of systems of $n$-variate
polynomial equations is specified through $n$ monomial bases. The natural locus
for the roots of those systems is known to be a certain toric variety. This
variety is a compactification of $(\mathbb C\setminus\{0\})^n$, dependent on
the monomial bases. A toric Newton operator is defined on that toric variety.
Smale's alpha theory is generalized to provide criteria of quadratic
convergence. Two condition numbers are defined and a higher derivative estimate
is obtained in this setting. The Newton operator and related condition numbers
turn out to be invariant through a group action related to the momentum map. A
homotopy algorithm is given, and is proved to terminate after a number of
Newton steps which is linear on the condition length of the lifted homotopy
path. This generalizes a result from Shub (2009).
