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

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.