Title: Multistationarity and Bistability for Fewnomial Chemical Reaction Networks,
Abstract: Bistability and multistationarity are properties of reaction networks linked
to switch-like responses and connected to cell memory and cell decision making.
Determining whether and when a network exhibits bistability is a hard and open
mathematical problem. One successful strategy consists of analyzing small
networks and deducing that some of the properties are preserved upon passage to
the full network. Motivated by this we study chemical reaction networks with
few chemical complexes. Under mass-action kinetics the steady states of these
networks are described by fewnomial systems, that is polynomial systems having
few distinct monomials. Such systems of polynomials are often studied in real
algebraic geometry by the use of Gale dual systems. Using this Gale duality we
give precise conditions in terms of the reaction rate constants for the number
and stability of the steady states of families of reaction networks with one
non-flow reaction.