Abstract. It is well known that various questions of stability of polynomial vectors fields can be reduced to quantifier elimination problems on real closed fields. More recently we have shown that also the parametric question of the occurrence of Hopf bifurcations can be decided by quantifier elimination. The combination of general purpose quantifier elimination systems has been sufficient to solve some of the occurring quantifier elimination problems but did not succeed for many others (on current computers). For the common case of equilibrium points with nonzero Jacobian determinant we will show that there is a computationally well suited description that can serve as an infrastructure for more efficient methods.