 # Consider the given system of polynomial equations, where all the coefficients are in <mi mathvari Jonathan Miles 2022-07-02 Answered
Consider the given system of polynomial equations, where all the coefficients are in $\mathbb{C}$:
$\left\{\begin{array}{l}{y}^{n}=P\left(x\right)\\ Q\left(x,y\right)=0\end{array}$
I would like to establish that either this system has solutions $\left(x,y\right)\in \mathbb{C}$ 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 $y$.
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You can determine the resultant $R\left(x\right)=Re{s}_{y}\left({y}^{n}-P\left(x\right),Q\left(x,y\right)\right)$ 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 $\left(x,y\right)$ a solution (of the common factor and thus of the system).
If the resultant is not zero, then $R\left(x\right)$ is a polynomial. Only at the roots of this polynomial you will then find at least one $y$ so that $\left(x,y\right)$ is a solution of the system.