Polynomial Mappings on Finite Commutative Rings

If R is a finite commutative ring then, unless R is a field, not every function on R can be represented by a polynomial with coefficients in R. We investigate how many functions on R are induced by polynomials in R[x]. In some cases, we can determine the structure of the semi-group of polynomial functions (with respect to composition) on R and of the group of permutations induced by polynomials. If the finite ring is a residue class ring of a Dedekind domain, we can answer a question of Narkiewicz and characterize those ideals I such that every function on D/I is induced by a polynomial in Int(D) (i.e., a polynomial with coefficients in the quotient field of D that maps D to itself), which preserves congruences mod I. Another object of investigation are permutation polynomials in several variables over a finite commutative ring. Here we could classify the rings for which the two different ways of defining the concept agree.