Symmetric rational function in two variables
= Symmetric rational function in two variables
Every rational function in $\mathbb C(X,Y)$ fixed by exchanging $X$ and $Y$ is a quotient of coprime symmetric polynomials. In a coprime presentation $p/q$, symmetry gives
$$
pq^*=p^*q.
$$
Unique factorization implies $p^*=\lambda p$ and $q^*=\lambda q$. The alternative $\lambda=-1$ would make both divisible by $X-Y$, contradicting coprimality, so $\lambda=1$.