  Much attention has been focused in recent years on the following algebraic
problem arising from applications: which chemical reaction networks, when taken
with mass-action kinetics, admit multiple positive steady states? The interest
behind this question is in steady states that are stable. As a step toward this
difficult question, here we address the question of multiple nondegenerate
positive steady states. Mathematically, this asks whether certain families of
parametrized, real, sparse polynomial systems ever admit multiple positive real
roots that are simple. Our main results settle this problem for certain types
of small networks, and our techniques point the way forward for larger
networks.
