KöMaL 2018 A718

Let $\mathbb{R}[x,y]$ denote the set of two-variable polynomials with real coefficients. We say that the pair $(a,b)$ is a zero of the polynomial $f\in\mathbb{R}[x,y]$ if $f(a,b)=0$. If polynomials $p,q\in\mathbb{R}[x,y]$ have infinitely many common zeros, does it follow that there exists a non-constant polynomial $r\in\mathbb{R}[x,y]$ which is a factor of both $p$ and $q$?