Lifting of cycles in functional graphs

For a given function from a set to itself, we can define a directed graph called the functional graph, where the vertices are the elements of the set, and the edges are all the pairs of inputs and outputs for the function. In this article we consider functional graphs on [Formula: see text] with respect to polynomial functions. The main result describes the behavior of cycles in functional graphs on [Formula: see text] while [Formula: see text] is increasing, where [Formula: see text] is a prime number.