  We present conditions which guarantee a parametrization of the set of
positive equilibria of a generalized mass-action system. Our main results state
that (i) if the underlying generalized chemical reaction network has an
effective deficiency of zero, then the set of positive equilibria coincides
with the parametrized set of complex-balanced equilibria and (ii) if the
network is weakly reversible and has a kinetic deficiency of zero, then the
equilibrium set is nonempty and has a positive, typically rational,
parametrization. Via the method of network translation, we apply our results to
classical mass-action systems studied in the biochemical literature, including
the EnvZ-OmpR and shuttled WNT signaling pathways. A parametrization of the set
of positive equilibria of a (generalized) mass-action system is often a
prerequisite for the study of multistationarity and allows an easy check for
the occurrence of absolute concentration robustness (ACR), as we demonstrate
for the EnvZ-OmpR pathway.
