  We show that the toric locus of a reaction network is a smoothly embedded
submanifold of the Euclidean space. More precisely, we prove that the toric
locus of a reaction network is the image of an embedding and it is
diffeomorphic to the product space between the affine invariant polyhedron of
the network and its set of complex-balanced flux vectors. Moreover, we prove
that within each affine invariant polyhedron, the complex-balanced equilibrium
depends smoothly on the parameters (i.e., reaction rate constants). We also
show that the complex-balanced equilibrium depends smoothly on the initial
conditions.
