Let
$\mathbb{K}=\overline{\mathbb{Q}}$ or $\mathbb{K}=\mathbb{C}$,
$R(x,y)\in\mathbb{K}(x,y)\setminus\mathbb{K}(x)\setminus\mathbb{K}(y)$.
How can we specify all types of $R(x,y)$ so that a $P(x,y)\in\mathbb{K}[x,y]$ without a univariate factor exists so that the equations $R(x,y)=0$ and $P(x,y)=0$ have the same solution set?
I need the answers for proving a new theorem about solving transcendental equations of one variable.
Let's take $R(x,y)$ in reduced form, that means, there are $p(x,y),q(x,y)\in\overline{\mathbb{K}}[x,y]$ coprime so that $\frac{p(x,y)}{q(x,y)}=R(x,y)$.
Until now, I can prove only that particular forms of $p(x,y),q(x,y)$ imply that $P(x,y)$ has a univariate factor.
But I need the opposite direction of proof.
Is a univariate-factor free $p(x,y)$ a necessary and sufficient condition? If yes, how can I prove that?