Polynomial Factorisation and an Application to Regular Directed Graphs

The main theme is the distribution of polynomials of given degree which split into a product of linear factors over a finite field. The work was motivated by the following problem on regular directed graphs. Extending a notion of Chung, Katz has defined a regular directed graph based on thek-algebrak[X]/(f), wherekis the finite field of orderqandfa monic polynomial of degreenoverk. It is shown that the diameter of this graph is at mostn+2 wheneverq≥B(n)=[n(n+2)!]2. This improves on the work of Katz who gave a similar result for square-free polynomialsfwithout specifyingB(n).