Let $f: E \subset \mathbb{R}^n \to \mathbb{R}^m$ such that $E$ is open connected set, and $D_{k+1} f(x) = 0$ for all $x \in E$ then $f$ is a polynomial in $x_1 ,x_2, \cdots ,x_n$ of degree at most $k$ ?
Similar proof we did for one-variable function, but the multi-variable thing here is the thing i am unable to handle