Characteristic functions for \MakeLowercase{(r, n)}-free and \MakeLowercase{(f, g)}-free elements

TL;DR

This paper develops characteristic functions for identifying (r, n)-free and (f, g)-free elements in finite fields and modules, providing a unified approach to their algebraic properties and distributions.

Contribution

It introduces a new characteristic function framework for (r, n)-free and (f, g)-free elements in finite cyclic modules and polynomial value contexts.

Findings

Derived explicit characteristic functions for (r, n)-free elements.

Extended the concept to (f, g)-free elements via polynomial evaluations.

Provided formulas applicable to finite fields and modules.

Abstract

For a prime power $q$, $\F$ denotes the finite field of order $q$, and for $m\geq 2$, $\Fm$ denotes the extension field of degree $m$. We establish a characteristic function for the set of $(r,\, n)$-free elements of finite cyclic $R$-module for the Euclidean domain $R$. Furthermore, we explore $(f,\, g)$-freeness through polynomial values and finally give an expression for the characteristic function for the set of $(f,\, g)$-free elements.