Lie-Poincare' transformations and a reduction criterion in Landau theory

In the Landau theory of phase transitions one considers an effective potential $\Phi$ whose symmetry group $G$ and degree $d$ depend on the system under consideration; generally speaking, $\Phi$ is the most general $G$-invariant polynomial of degree $d$. When such a $\Phi$ turns out to be too complicate for a direct analysis, it is essential to be able to drop unessential terms, i.e. to apply a simplifying criterion. Criteria based on singularity theory exist and have a rigorous foundation, but are often very difficult to apply in practice. Here we consider a simplifying criterion (as stated by Gufan) and rigorously justify it on the basis of classical Lie-Poincar\'e theory as far as one deals with fixed values of the control parameter(s) in the Landau potential; when one considers a range of values, in particular near a phase transition, the criterion has to be accordingly partially modified, as we discuss. We consider some specific cases of group $G$ as examples, and study in detail the application to the Sergienko-Gufan-Urazhdin model for highly piezoelectric perovskites.