  We establish how the coefficients of a sparse polynomial system influence the
sum (or the trace) of its zeros. As an application, we develop numerical tests
for verifying whether a set of solutions to a sparse system is complete. These
algorithms extend the classical trace test in numerical algebraic geometry. Our
results rely on both the analysis of the structure of sparse resultants as well
as an extension of Esterov's results on monodromy groups of sparse systems.
