I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all over the world,and hope that I can get a complete answer or some constructive suggestions.

Given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}\text{, and }\frac{P_3(y_1,y_2,\dots,y_n)}{P_4(y_1,y_2,\dots,y_n)}$$ where $P_i$ is polynomial(that is $P_1(x_1,x_2,\dots,x_n), P_2(x_1,x_2,\dots,x_n), P_3(y_1,y_2,\dots,y_n), P_4(y_1,y_2,\dots,y_n)$ are polynomial) whose coefficients are over $\mathbb{Q}$,let $J$ be Jacobian matrix, when does $f(x_1,x_2,\dots,x_n)$ exist,such that $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}=\frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}|\det (J(f(x_1,x_2,\dots,x_n)))|\text{ ,}$$ and $f(x_1,x_2,\dots,x_n)$ is bijective ?

When $f(x_1,x_2,\dots,x_n)$ is restricted to differential polynomials, rational functions, algebraic functions,or just all differential function including transcendental functions over $\mathbb{C}$ , $\mathbb{R}$, what is the condition or non-condition for the existence of $f(x_1,x_2,\dots,x_n)$ ?