TITLE: Two polynomial problem
QUESTION [5 upvotes]: Suppose that there is a path $g(t) = (p(t), q(t))$ in a plane where $p$ and $q$ are polynomials. Is there always a polynomial $f(x,y)$ different from zero polynomial such that the image of $g$ is contained in the set $\{(x,y) | f(x,y) = 0\}$?

REPLY [2 votes]: I do not resist to give an elementary solution. Let $n,m$ the degrees of the polynomials $p,q$. Let $K$ an integer. There is a priori $(K+1)^2$ polynomials of the form $p^k q^l$ with $0\leq k\leq K$ and $0\leq l\leq K$. But the degree of such polynomial is $kn+lm\leq K(n+m)$. So they belongs to a vector space of dimension $K(n+m)+1$. For large $K$, we have $(K+1)^2>K(n+m)+1$, and the monomials $p^kq^l$ must be dependant over the base field, so there exists $f(x,y)$ not $0$ such that $f(p(t),q(t))=0$.