Consider two [multilinear](https://en.wikipedia.org/wiki/Multilinear_polynomial) 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?



Maximizing the ratio of multilinear polynomials
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 $x\_3$.


