  A multistationarity region is the part of a reaction network's parameter
space that gives rise to multiple steady states. Mathematically, this region
consists of the positive parameters for which a parametrized family of
polynomial equations admits two or more positive roots. Much recent work has
focused on analyzing multistationarity regions of biologically significant
reaction networks and determining whether such regions are connected; indeed, a
better understanding of the topology and geometry of such regions may help
elucidate how robust multistationarity is to perturbations. Here we focus on
the multistationarity regions of small networks, those with few species and few
reactions. For two families of such networks -- those with one species and up
to three reactions, and those with two species and up to two reactions -- we
prove that the resulting multistationarity regions are connected. We also give
an example of a network with one species and six reactions for which the
multistationarity region is disconnected. Our proofs rely on the formula for
the discriminant of a trinomial, a classification of small multistationary
networks, and a recent result of Feliu and Telek that partially generalizes
Descartes' rule of signs.
