Let $K$ be a compact subset of $\mathbb{R}^2$. Let $P$ be a polynomial in the variables $x$ and $y$. Given $\epsilon>0$, can we find two polynomials $P_1=P_1(x)$ and $P_2=P_2(y)$ such that $ \sup_{x,y\in K}|P(x,y) - (P_1(x)+P_2(y))| \leq \epsilon ?$
My feeling is No, but I don't see how to prove it...
EDIT : Concerning my feeling, I had in mind the case when $K$ is the unit disc. As mentioned by Nate Eldredge in the comment below, the question depends on $K$. I consequently ask the question assuming that there exists no relation like $y=f(x)$ parametrizing $K$, where $f$ is continuous where $x$ runs.