On polynomial functions from Zn1 × Zn2 × … × Znr to Zm Original Research Article

The well-known concept of a polynomial function (mod m) has been generalized to polynomial functions from Zn to Zm, and a number of results have been obtained in (Chen, 1995). In the present paper, we further define the concept of polynomial functions from Zn1 × Zn2 × … × Znr to Zm, 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, …, nr and m for all functions from Zn1 × Zn2 × … × Znr to Zm to be polynomial functions. Further, we give an answer to the following problem: How to determine whether a given function from Zn1 × Zn2 × … × Znr to Zm is a polynomial function, and how to obtain a polynomial to represent a polynomial function from Zn1 × Zn2 × … × Znr to Zm?