  An important invariant of a chemical reaction network is its maximum number
of positive steady states. This number, however, is in general difficult to
compute. Nonetheless, there is an upper bound on this number -- namely, a
network's mixed volume -- that is easy to compute. Moreover, recent work has
shown that, for certain biological signaling networks, the mixed volume does
not greatly exceed the maximum number of positive steady states. Continuing
this line of research, we further investigate this overcount and also compute
the mixed volumes of small networks, those with only a few species or
reactions.
