Polynomial dynamical systems are widely used to model and study real
phenomena. In biochemistry, they are the preferred choice for modelling the
concentration of chemical species in reaction networks with mass-action
kinetics. These systems are typically parameterised by many (unknown)
parameters. A goal is to understand how properties of the dynamical systems
depend on the parameters. Qualitative properties relating to the behaviour of a
dynamical system are locally inferred from the system at steady state. Here we
focus on steady states that are the positive solutions to a parameterised
system of generalised polynomial equations. In recent years, methods from
computational algebra have been developed to understand these solutions, but
our knowledge is limited: for example, we cannot efficiently decide how many
positive solutions the system has as a function of the parameters. Even
deciding whether there is one or more solutions is non-trivial. We present a
new method, based on so-called injectivity, to preclude or assert that multiple
positive solutions exist. The results apply to generalised polynomials and
variables can be restricted to the linear, parameter-independent first
integrals of the dynamical system. The method has been tested in a wide range
of systems.Comment: Final version, Proceedings of the Royal Society
