Congruence preserving functions in the residue class rings of polynomials over finite fields

TL;DR

This paper studies functions that preserve congruences in polynomial residue rings over finite fields, providing formulas, conditions, and characterizations for these functions, extending classical integer results to polynomial rings.

Contribution

It introduces the concept of congruence preserving functions in polynomial residue rings over finite fields and characterizes when these functions are also polynomial functions.

Findings

Derived a counting formula for congruence preserving functions

Identified conditions under which all such functions are polynomial functions

Characterized the structure of these functions in the polynomial residue context

Abstract

In this paper, as an analogue of the integer case, we define congruence preserving functions over the residue class rings of polynomials over finite fields. We establish a counting formula for such congruence preserving functions, determine a necessary and sufficient condition under which all congruence preserving functions are also polynomial functions, and characterize such functions.