# Polynomial functions in the residue class rings of Dedekind domains

## TL;DR

This paper extends the concept of polynomial functions to residue class rings of Dedekind domains, providing canonical forms and counting formulas, including explicit results for polynomial rings over finite fields.

## Contribution

It introduces a general framework for polynomial functions over Dedekind domain residue rings and derives explicit counting formulas, advancing the algebraic understanding of these functions.

## Key findings

- Canonical representations for polynomial functions over Dedekind domain residue rings
- Counting formulas for such polynomial functions
- Explicit enumeration over polynomial rings of finite fields

## Abstract

In this paper, as an extension of the integer case, we define polynomial functions over the residue class rings of Dedekind domains, and then we give canonical representations and counting formulas for such polynomial functions. In particular, we give an explicit formula for the number of polynomial functions over the residue class rings of polynomials over finite fields.