Maximizing the ratio of multilinear polynomials

Question

Consider two multilinear Polynomials $A(x_1,x_2,x_3,\dotsc,x_n)$ and $B(x_1,x_2,x_3,\dotsc,x_n)$ of $n > 2$ variables $x_i \in \mathbb{R}$ and their ratio \begin{equation*} F(x_1,x_2,x_3,\dotsc,x_n) = \frac{A(x_1,x_2,x_3,\dotsc,x_n)}{B(x_1,x_2,x_3,\dotsc,x_n)} \end{equation*}

Define $G$ to be the function obtained from $F$ by substituting $(x_1,x_2)=(t,-t)$ and maximizing over $t$: \begin{equation*} G(x_3,\dotsc,x_n) := \max_t F(t,-t,x_3,\dotsc,x_n) \end{equation*}

Here the maximum is assumed to exist in a local sense, i.e. assume there is a function $t = f(x_3,\dotsc,x_n)$ that locally parametrizes the solution set of \begin{equation} \frac{\partial}{\partial t} F(t,-t,x_3,\dotsc, x_n)=0. \end{equation} such that we have \begin{equation} G(x_3,\dotsc,x_n) := F(f(x_3,\dotsc,x_n),-f(x_3,\dotsc,x_n),x_3,\dotsc, x_n) \end{equation}

Question: Can $G$ be written as \begin{equation*} G(x_3,\dotsc,x_n) = \frac{\widetilde{A}(x_3,\dotsc,x_n)}{\widetilde{B}(x_3,\dotsc,x_n)} \end{equation*} with multilinear polynomials $\widetilde{A},\widetilde{B}$ of the remaining $n-2$ variables?

Answer 1

Not in general. E.g., let $n=3$, $A(x_1,x_2,x_3):=1-x_1 x_2+2 x_3 x_2+x_1 x_3$, and $B(x_1,x_2,x_3):=2-x_1 x_2+x_3 x_2+2 x_1 x_3$. Then for $x_3\in(0,2\sqrt2)$ $$G(x_3)=\max_{t\in\mathbb R}\frac{A(t,-t,x_3)}{B(t,-t,x_3)} =\frac{x_3^2+2 \sqrt{6 x_3^2+1}+6}{8-x_3^2},$$ which is not the ratio of multilinear polynomials in $x_3$.