Let $u, v \in \mathbb{R}[x,y]$ satisfying $u_{x} = v_{y}$ and $u_{y} = -v_{x}$ everywhere in $\mathbb{C}$. Is the function $f(x + iy) = u(x,y) + iv(x,y)$ a polynomial in the variable $z = x + iy$?
I really don't know where to start with this. I tried to build a counterexample, but I had no success.
Assuming that (as in the title) $u$ and $v$ are polynomial functions, then yes, it is true. The function $f$ is holomorphic and therefore analytic. And, since $u$ and $v$ are polynomial functions $f^{(n)}=0$ is $n$ is large enough. Therefore, $f$ is polynomial too.