  We discuss parameter dependent polynomial ordinary differential equations
that model chemical reaction networks. By classical quasi-steady state (QSS)
reduction we understand the following familiar heuristic: Set the rate of
change for certain (a priori chosen) variables equal to zero and use the
resulting algebraic equations to obtain a system of smaller dimension for the
remaining variables. This procedure will generally be valid only for certain
parameter ranges. We start by showing that the reduction is accurate if and
only if the corresponding parameter is what we call a QSS parameter value, and
that the reduction is approximately accurate if and only if the corresponding
parameter is close to a QSS parameter value. These QSS parameter values can be
characterized by polynomial equations and inequations, hence parameter ranges
for which QSS reduction is valid are accessible in an algorithmic manner. A
closer investigation of QSS parameter values and the associated systems shows
the existence of certain invariant sets; here singular perturbations enter the
picture in a natural manner. We compare QSS reduction and singular perturbation
reduction, and show that, while they do not agree in general, they do, up to
lowest order in a small parameter, for a quite large and relevant class of
examples. This observation, in turn, allows the computation of QSS reductions
even in cases where an explicit resolution of the polynomial equations is not
possible.

Classical Analysis and ODEs;Dynamical Systems