We discuss parameter dependent polynomial ordinary differential equations that model chemical reaction networks. By classical quasisteady state (QSS) reduction we understand the following familiar (heuristically motivated) mathematical procedure: 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. The 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 defining characteristic of a QSS parameter value is that the algebraic variety * Corresponding author. Email walcher@matha.rwth-aachen.de, Phone +49 241 809 8132, Fax +49 241 809 2212. defined by the QSS relations is invariant for the differential equation.A closer investigation of the associated systems shows the existence of further 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. MSC (2010): 92C45, 34E15, 80A30, 13P10

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There is a systematic approach to the computation of quasi-steady state reductions, employing the classical theory of Tikhonov and Fenichel, rather than the commonly used ad-hoc method. In the present paper we discuss the relevant case that the local slow manifold (in the asymptotic limit) is a vector subspace, give closed-form expressions for the reduction and compare these to the ones obtained by the customary method. As it turns out, investment of more theory pays off in the form of simpler reduced systems. Applications include a number of standard models for reactions in biochemistry, for which the reductions are extended to the fully reversible setting. In a short final section we illustrate by example that a QSS assumption may be erroneous if the hypotheses for Tikhonov's theorem are not satisfied.