Generalizations of Euler's $\varphi$-function with respect to systems of polynomials of several variables

TL;DR

This paper introduces a new generalization of Euler's -function linked to multivariable polynomial systems, rederives known identities, explores novel special cases, and discusses their unitary analogues.

Contribution

It presents a novel generalization of Euler's -function for polynomial systems, offering new identities and special cases not previously documented.

Findings

Reproved known identities with a short direct approach

Studied new special cases of the generalized -function

Discussed unitary analogues of these functions

Abstract

We introduce a new generalization of Euler's $\varphi$-function associated with a system of polynomials of several variables. We reprove by a short direct approach certain known related identities, and study some other special cases that do not appear in the literature. We also discuss the unitary analogues of these functions.