Let $f \in \mathbb{C}[z_{1}, . . . , z_{n}]$ be a nonzero polynomial ($n ≥ 1$) and
$X = \{ z ∈ \mathbb{C}^n| f(z) = 0 \}.$
How do we prove that $\mathbb{C}^n\setminus X$ is path connected?
In one variable case the polynomial has only finitely many roots and by considering straight lines passing through any two distict points we can conclude that $\mathbb{C}^n \setminus X$ is path connected. How do we proceed for any $n$?
Best Answer
Hint: Any two points $a, b \in \mathbb{C}^n$ lie on the "line" $\{a + (b -a) z \mid z \in \mathbb{C}\}$. Use this to reduce to the one-variable case.