On the structure of digraphs of polynomial transformations over finite commutative rings with unity

The paper describes structural characteristics of the digraph of an arbitrary polynomial transformation of a finite commutative ring with unity. A classification of vertices of the digraph is proposed: cyclic elements, initial elements, and branch points are described. Quantitative results on such objects and heights of vertices are given. Besides, polynomial transformations are shown to have cycles whose lengths coincide with the lengths of cycles of the induced polynomial transformation over the field $R/\Re$, where $\Re$ is the radical of the finite commutative local ring $R$.