This paper regroups various studies achieved around polynomial dynamical system theory. It presents the basic algebraic tools for the study of this particular class of discrete event systems. The polynomial dynamical systems are defined by polynomial equations over Z/3Z. Their study relies on concept borrowed from elementary algebraic geometry: varieties, ideals and morphisms. They are the basic tools that allow us to translate properties or specifications from a geometric description to suitable polynomial computations. In this paper, we more precisely describe the controller synthesis methodology. We specify the main requirements as simple properties, named control objectives, that the controlled plant has to satisfy.The plant is specified as a polynomial dynamical system over Z/3Z. The control of the plant is performed by restricting the controllable input values to values suitable with respect to the control objectives. This restriction is obtained by incorporating new algebraic equations into the initial polynomial dynamical system, which specifies the plant. Various kind of control objectives are considered, such as ensuring the invariance or the reachability of a given set of states, as well as partial order relation to be checked by the controlled plant.