## Abstract

We introduce the notion of a P-ordering of an arbitrary subset X of a Dedekind ring R, and use it to investigate the functions from X to R which can be represented by polynomials. In the case when R is a finite principal ideal ring, our results include canonical representations for those polynomials which vanish on X, a canonical representation for each polynomial function from X to R, necessary and sufficient congruence conditions for a function from X to R to be a polynomial function, and a formula for the number of such functions. When R is a Dedekind domain, we use the concept of P-ordering to give necessary and sufficient conditions for the existence of a regular basis for the ring Int(X, R), and moreover, we give an explicit construction of such a basis whenever it exists. Finally, we deduce many of the classical theorems of Kempner, Carlitz, Pólya, and others on polynomial mappings as special cases of our results.

Original language | English (US) |
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Pages (from-to) | 101-127 |

Number of pages | 27 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Volume | 490 |

State | Published - Dec 1 1997 |

## All Science Journal Classification (ASJC) codes

- General Mathematics
- Applied Mathematics