Let $f$ be a function on $R^2$ that is a polynomial in $y$ when $x$ is fixed and that is a polynomial in $x$ when $y$ is fixed. Is it true that $f$ is a polynomial?
The only idea I come up with is the characterization of one-variable polynomials. A function $p(x)$ is a polynomnial iff $p^{(n)}(x)=0$ for some $n$. This gives that the $m$th partials of $f$ are zero for $m \in \mathbb Z_+$. But how to conclude that $f$ is a polynomial in two variables? Is there any criterion that I can use?