Polynomial Mappings on Finite Commutative Rings

    Project: Research project

    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.
    StatusFinished
    Effective start/end date1/01/9331/01/07