### Description

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.

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.

Status | Finished |
---|---|

Effective start/end date | 1/01/93 → 31/01/07 |