Is there a special name for morphisms sharing some special condition?

Let $\Phi=[F_1,\ldots,F_N]$ be a map, with $F_i$ homogenous polynomials of variables $X_1,\ldots,X_N$ of the same degree with integer coefficients.

The map $\Phi$ mapping $Z^N$ into itself may share the following property:

Let $P=(x_1,\ldots,x_N)\in Z^N$ be any point such that $\gcd (x_1,\ldots,x_N)=1$, then $\gcd$ of all coordinates of $\Phi(P)$ equals $1$ as well.

For example $\Phi=[X_1^2,2X_1X_2-X_2^2]$ satisfies this property.

Has this property been studied or even more is there a name for it?