Consider the given system of polynomial equations, where all the coefficients are in $\mathbb{C}$:

$\{\begin{array}{l}{y}^{n}=P(x)\\ Q(x,y)=0\end{array}$

I would like to establish that either this system has solutions $(x,y)\in \mathbb{C}$ for all $x\in \mathbb{C}$, or either has solutions for countably (or even better, finitely) many $x\in \mathbb{C}$. I am not completely sure this is true but haven't found any counterexamples yet.

So far, I have found that if we have solutions for infinitely many $x$, then we have solutions for infinitely many $y$.

$\{\begin{array}{l}{y}^{n}=P(x)\\ Q(x,y)=0\end{array}$

I would like to establish that either this system has solutions $(x,y)\in \mathbb{C}$ for all $x\in \mathbb{C}$, or either has solutions for countably (or even better, finitely) many $x\in \mathbb{C}$. I am not completely sure this is true but haven't found any counterexamples yet.

So far, I have found that if we have solutions for infinitely many $x$, then we have solutions for infinitely many $y$.