$p$ is a homogeneous polynomial. Show that $p^{-1}(a)$ is a submanifold of $\mathbb{R}^n$ if $a\neq 0$. The submanifolds of $a>0$ are diffeomorphic.

Question

Let $p$ be a homogeneous polynomial of degree $m$ in $n$ variables $t_1,\cdots, t_n$. Show that for each $a\neq 0$, $p^{-1}(a)$ is a submanifold of codimension 1 in $\mathbb{R}^n$. Show that the submanifolds obtained with $a>0$ are all diffeomorfic, as well as those with $a<0$.

Using Euler's identity one can see that, for each $t\in p^{-1}(a)$, $$\sum_{i=1}^n t_i\frac{\partial p}{\partial t_i} (t)=ma.$$ So, for some $j\in\{1,\cdots,n\}$, $\frac{\partial p}{\partial t_i} (t)\neq 0$. So $a$ is a regular value and, therefore, $p^{-1}(a)$ is a embedded submanifold of $\mathbb{R}^n$ of dimensions $n-1$.

How do I show the second statement?

Answer 1

The thing about being homogeneous of degree $m$ is that $p(tx) = t^mp(x)$ for $t >0$. Using this should give you an idea for an explicit diffeomorphism from $p^{-1}(t^ma)$ to $p^{-1}(a)$.

Answer 2

It is straightforward to see that $p^{-1}(a)$ and $p^{-1}(-a)$ are diffeomorphic, so it suffices to show that for $0<a<b$, the level sets $p^{-1}(a)$ and $p^{-1}(b)$ are diffeomorphic, but as you noticed $0$ in the only possible critical value of $p$. In particular, there is no critical value of $p$ between $a$ and $b$ and the statement follow from the standard critical non-crossing theorem in Morse theory.