Can we choose constants such that a system of two polynomials in two variables has finitely many solutions? Given $f(x,y)$ and $g(x,y)$ be polynomials with rational coefficients does there exist $h, k \in \mathbb{R}$ such that the system:
$$
f(x,y) = h
$$
$$
g(x,y) = k
$$
has finitely many solutions $(x,y)$.
My attempt
Using Sard's theorem, we know that if we correspond the solutions $(x,y)$ to critical values of some map $\phi$ then we will have only finitely many solutions. I'm struggling to understand what values I should be giving to $h$ and $k$ here, as well as what $\phi$ would look like.

The answer depends on the polynomials $f$ and $g$ that you start with. For example, if $f(x,y)=x$ and $g(x,y)=y$ then for any $h,k$ you will have unique solution. On the other hand, if $f(x,y)$ is a multiple of $g(x,y)$ then you may not choose any $h,k$ for the system to have finitely many solutions (if no solutions doesn't mean finitely many solutions).
A useful theorem that you can apply here is the Bezout's theorem, which says that if $f(x,y)-h$ and $g(x,y)-k$ don't have common factor then they will have finitely many solutions or no solutions.
