First question: given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}=\frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}|\det (J(f(x_1,x_2,\dots,x_n)))|$$ where $P_i$ is polynomial(that is $P_3(y_1,y_2,\dots,y_n)$,$P_4(y_1,y_2,\dots,y_n)$ are polynomial,but we are not sure if $P_3(f(x_1,x_2,\dots,x_n)),P_4(f(x_1,x_2,\dots,x_n))$ are polynomial,since $f(x_1,x_2,\dots,x_n))$ may not be polynomial,or its' Jacobian determinant may not be polynomial) whose coefficients are over $\mathbb{Q}$,$J$ is the Jacobian matrix, does there always exist $f(x_1,x_2,\dots,x_n)$ that is a polynomial? If the polynomial exists,then is it injective?If the polynomial does not exists,then does there exist injective function $f(x_1,x_2,\dots,x_n)$ and what class of function does $f(x_1,x_2,\dots,x_n)$ lie in?
Second question: Suppose $ f(x_1,x_2,\dots,x_n)$ is polynomial,does there exist functions $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}\text{,and }\frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}$$ such that $$\forall f(x_1,x_2,\dots,x_n),\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}\neq \frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}|\det (J(f(x_1,x_2,\dots,x_n)))|$$ If such functions exist,is there a function $\psi(x_1,x_2,\dots,x_n)$ such that $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}=\frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}|\det (J(\psi(x_1,x_2,\dots,x_n)))|\text{ ?}$$ $\text{here $\psi(x_1,x_2,\dots,x_n)$ is not a polynomial.}$
Please pay attention to $$f(x_1,x_2,\dots,x_n)$$ in First question and Second question,in first question $f(x_1,x_2,\dots,x_n)$ is just a function,but in second question $f(x_1,x_2,\dots,x_n)$ is supposed to be a polynomial,and $\psi(x_1,x_2,\dots,x_n)$ is supposed not to be a polynomial.
I am not sure if they are a special case of Jacobian conjecture. I ask the questions on mathoverflow again,here , since I have not gotten a answer or a comment.