  The quasi-steady-state assumption (QSSA) is an approximation that is widely
used in chemistry and chemical engineering to simplify reaction mechanisms. The
key step in the method requires a solution by radicals of a system of
multivariate polynomials. However, Pantea, Gupta, Rawlings, and Craciun showed
that there exist mechanisms for which the associated polynomials are not
solvable by radicals, and hence reduction by QSSA is not possible. In practice,
however, reduction by QSSA always succeeds. To provide some explanation for
this phenomenon, we prove that solvability is guaranteed for a class of common
chemical reaction networks. In the course of establishing this result, we
examine the question of when it is possible to ensure that there are finitely
many (quasi) steady states. We also apply our results to several examples, in
particular demonstrating the minimality of the nonsolvable example presented by
Pantea, Gupta, Rawlings, and Craciun.
