Problem: Let $P: \mathbb{R}^{n+1} \ \lbrace 0 \rbrace \rightarrow \mathbb{R}^{k+1} \ \lbrace 0 \rbrace$ be a smooth function and suppose that $P(\lambda x)=\lambda^d P(x) \ \forall \ \lambda \in \mathbb{R} \ \lbrace 0 \rbrace, d \in \mathbb{Z}$. Show that the map $\tilde{P} : \mathbb{R}\mathbb{P}^n \rightarrow \mathbb{R}\mathbb{P}^k$ defined by $\tilde{P}([x])=[P(\lambda x)]$ is well-defined and smooth.
I can show that $\tilde{P}$ is well-defined easily enough. It is the smoothness part that bothers me. My idea was initially to use that $P$ is smooth and work with natural maps $\pi : \mathbb{R}^{n+1} \ \lbrace 0 \rbrace \rightarrow \mathbb{R}\mathbb{P}^n$ and the charts used to prove that $\mathbb{R}\mathbb{P}^n$ is a smooth manifold. But I haven't gotten any further. If anyone has a hint, I would like to hear it.
Here's a hint: Since the map is well-defined, and smoothness is a local condition, it's enough to check smoothness in some chart. In the natural charts, $\mathbb{RP}^n$ and $\mathbb{RP}^k$ look like $\mathbb{R}^{n}$ and $\mathbb{R}^{k}$, respectively, as collapsed subspaces of $\mathbb{R}^{n+1}$ and $\mathbb{R}^{k+1}$. What does $P$ look like in these charts? How does this relate to the smoothness you're given?