I am trying to solve the following problem for my introductory course in Differential Analysis and Geometry.
$M = \{[z]\in \mathbb{C}P^n:\sum_{i=0}^{n+1}z_i^m=0\}$ is an embedded submanifold of the complex projective space. I want to try the regular value theorem but I do not know how I can show that $0$ is a regular value. Where we have $m$ an integer. Thanks in advance.
HINT: Either work in the ($n+1$) standard coordinate charts on $\Bbb CP^n$ or ... for future reference ... prove the following lemma:
If $f$ is a homogeneous polynomial on $\Bbb C^{n+1}$, then $0$ is a regular value of $f$ on $\Bbb C^{n+1}-\{0\}$ if and only if $0$ is a regular value of $F(z_1,\dots,z_n) = f(1,z_1/z_0,\dots,z_n/z_0)$.