Every function from a finite field to itself can be represented by
a polynomial. The functions which are also permutations give
rise to ``permutation polynomials," which have potential
applications in cryptology. We will introduce a generalization of
permutation polynomials called ``degree-preserving polynomials"
and show a classification scheme of the latter.
The criteria for a polynomial to qualify as degree preserving are
certainly less stringent than those for the permuting
qualification. Thus the idea to study degree-preserving
polynomials allows more
opportunity to maneuver and gain intuition about the occurrence of
such polynomials.