Congruence preservation and polynomial functions from ℤ_n to ℤ_m

A function f: ℤ_n → ℤ_m 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 ℤ_n to ℤ_m 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 : ℤ_n → ℤ_m necessarily a polynomial function? In the present paper, we give a complete answer to Chen's question by determining all such pairs. We also show that 7/π² is the natural density in ℕ of the set of all m such that the polynomial functions from ℤ_m to itself are precisely the congruence preserving functions from ℤ_m to itself. Moreover, we obtain a formula for the number of congruence preserving functions from ℤ_n to ℤ_m.