Congruence preservation and polynomial functions from

A function f: Zn → Zm is said to be congruence preserving if for all d dividing m and a, b ∈ {0, 1, …, n − 1}, a  b (mod d) implies f(a)  f(b) (mod d). In previous work, Chen defines the notion of a polynomial function from Zn to Zm and shows that any such function must also be a congruence preserving function. Chen then raises the question as to when the converse is true, i.e. for what pairs (n, m) is a congruence preserving function f : Zn → Zm necessarily a polynomial function? In the present paper, we give a complete answer to Chenʹs question by determining all such pairs. We also show that 7π2 is the natural density in N of the set of all m such that the polynomial functions from Zm to itself are precisely the congruence preserving functions from Zm to itself. Moreover, we obtain a formula for the number of congruence preserving functions from Zn to Zm.