Characterizing congruence preserving functions $Z/nZ\to Z/mZ$ via rational polynomials

TL;DR

This paper introduces a basis of rational polynomial-like functions for the set of functions from Z/nZ to Z/mZ and characterizes the subset of congruence preserving functions using linear combinations of these basis functions.

Contribution

It provides a novel basis for functions between modular rings and characterizes congruence preserving functions explicitly in terms of this basis.

Findings

The basis consists of functions P_0,...,P_{n-1} for functions Z/nZ to Z/mZ.

Congruence preserving functions are linear combinations of lcm(k)*P_k.

When n ≥ m, the number of congruence preserving functions is independent of n.

Abstract

We introduce a basis of rational polynomial-like functions $P_0,\ldots,P_{n-1}$ for the free module of functions $Z/nZ\to Z/mZ$. We then characterize the subfamily of congruence preserving functions as the set of linear combinations of the functions $lcm(k)\,P_k$ where $lcm(k)$ is the least common multiple of $2,\ldots,k$ (viewed in $Z/mZ$). As a consequence, when $n\geq m$, the number of such functions is independent of $n$.