On polynomial functions from Z_n1 × Z_n2 × ⋯ × Z_nr to Z_m

The well-known concept of a polynomial function (mod m) has been generalized to polynomial functions from Z_n to Z_m, and a number of results have been obtained in (Chen, 1995). In the present paper, we further define the concept of polynomial functions from Z_n1 × Z_n2 × ⋯ × Z_nr to Z_m and generalize the results of (Chen, 1995). We give a canonical representation and the counting formula for such polynomial functions. Then we obtain a necessary and sufficient condition on n₁,n₂, ... , n_r and m for all functions from Z_n1 × Z_n2 × ⋯ × Z_nr to Z_m to be polynomial functions. Further, we give an answer to the following problem: How to determine whether a given function from Z_n1 × Z_n2 × ⋯ × Z_nr to Z_m is a polynomial function, and how to obtain a polynomial to represent a polynomial function from Z_n1 × Z_n2 × ⋯ × Z_nr to Z_m?