Finding a polynomial $F$ such that $F(P,Q) = 0$, where $P$ and $Q$ are polynomials.

Question

I am a bit struggling with this at the moment :

Let $K$ be a field and let $P, Q \in K[X]$.
Is there always a (minimal?) polynomial $F \in K[Y,Z]$ such that $F(P,Q) = 0$?
And if/when the answer is positive, how to find such a polynomial?

For example, with $P = X+1$ and $Q = X^2$, we would have $F = Y^2 -2Y+ Z +1$ : $F(P,Q) = P^2 - 2P + Q + 1 = (X+1)^2 -2(X+1) + X^2 + 1 = 0$

I thank you in advance!

Answer 1

The resultant of $Y - P(X)$ and $Z - Q(X)$ with respect to $X$ is a polynomial $F(Y,Z)$ with the desired property.