  Dynamical systems arising from chemical reaction networks with mass action
kinetics are the subject of chemical reaction network theory (CRNT). In
particular, this theory provides statements about uniqueness, existence, and
stability of positive steady states for all rate constants and initial
conditions. In terms of the corresponding polynomial equations, the results
guarantee uniqueness and existence of positive solutions for all positive
parameters.
  We address a recent extension of CRNT, called generalized mass-action
systems, where reaction rates are allowed to be power-laws in the
concentrations. In particular, the (real) kinetic orders can differ from the
(integer) stoichiometric coefficients. As with mass-action kinetics, complex
balancing equilibria are determined by the graph Laplacian of the underlying
network and can be characterized by binomial equations and parametrized by
monomials. In algebraic terms, we focus on a constructive characterization of
positive solutions of polynomial equations with real and symbolic exponents.
  Uniqueness and existence for all rate constants and initial conditions
additionally depend on sign vectors of the stoichiometric and kinetic-order
subspaces. This leads to a generalization of Birch's theorem, which is robust
with respect to certain perturbations in the exponents. In this context, we
discuss the occurrence of multiple complex balancing equilibria. We illustrate
our results by a running example and provide a MAPLE worksheet with
implementations of all algorithmic methods.
