Abstract Lyapunov functions are a widely used tool to evaluate stability properties of nonlinear dynamical systems' equilibria. In this paper quantifier elimination is used to construct Lyapunov functions for polynomial systems from a parametric ansatz. Because the existing algorithms for quantifier elimination are inherently computationally expensive, a strategy is to simplify the quantifier elimination problem beforehand. A method is presented that may help simplifying the problem of finding suitable Lyapunov functions by deriving easier to evaluate necessary conditions. Finally, the introduced method is applied to an example system to show local asymptotic stability of an equilibrium point by constructing a Lyapunov function.