Consider the given system of polynomial equations, where all the coefficients are in $\mathbb{C}$: $$\begin{cases} y^n=P(x)\\ Q(x,y)=0\end{cases}$$ 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 $y$.
Any ideas?
Best Answer
You can determine the resultant $R(x)=Res_y(y^n-P(x),Q(x,y))$ of both polynomials eliminating $y$. If it is the zero polynomial, then the polynomials have a common factor, which means that for any $x$ you find at least one $y$ so that $(x,y)$ a solution (of the common factor and thus of the system).
If the resultant is not zero, then $R(x)$ is a polynomial. Only at the roots of this polynomial you will then find at least one $y$ so that $(x,y)$ is a solution of the system.