Given a rational function $R\in\Bbb R(x_1,\dots,x_n)$ with multilinear numerator and denominator, is there always a rational function $G\in\Bbb R(x)$ such that $G\circ R\in\Bbb R[x_1,\dots,x_n]$, $G(0)=0$, $G(1)=1$? In situations where such $G$ exists, what is the minimal degree of rational function $G\in\Bbb R(x)$ such that $G\circ R\in\Bbb R[x_1,\dots,x_n]$, $G(0)=0$, $G(1)=1$ where degree of rational function is maximum of degrees of numerator and denominator in lowest forms?

Since $G(0)=0,G(1)=1$, $G(x)=x\frac{C(x)}{D(x)}=x\frac{(x-1)A(x)+r}{(x-1)B(x)+r}$ holds with $r\neq0$.

By multilinear, I mean only monomials of form $ax_{i_1}x_{i_2}x_{i_3}\dots x_{i_d}$ with $i_j\in\{1,\dots,n\}$, $a\in\Bbb R$ are present.