Genericity Conditions for Finite Cyclicity of Elementary Graphics

In this paper we explore the Khovanskii method for proving the finite cyclicity of elementary graphics and how it can be applied in practice. The genericity conditions needed in that case form a proper subset of the usual methods. Moreover some of the conditions are non-intrinsic and can be artificially created by action of the automorphism groups preserving the normal forms near the singularities and their action on regular transitions. Hence we introduce an extension of the method which treats the usual functional-Pfaffian systems together with the admissible changes of coordinates in the functional equations.