Why is the set of points where a complex polynomial does not vanish connected?

Question

Let $p$ be a complex multivariate polynomial. Let $C$ be the set of those complex tuples where $p$ is nonzero. Then, $C$ is connected.

Answer 1

You may assume without loss of generality that $p\not\equiv 0$, since otherwise $C = \varnothing$ is clearly connected.

Let $z\in C$ be any point. We will show $C$ is connected by proving that for any point $w\in C$, there is a path in $C$ between $w$ and $z$. Indeed, let $L\cong \mathbb{C}$ be the complex line connecting $z$ and $w$. Then $p|_L$ is a complex polynomial in one variable that does not vanish everywhere on $L$ (since it does not vanish at either $z$ or $w$), and hence $p$ vanishes at at most finitely many points of $L$. Thus $L\cap C$ is $\mathbb{C}$ with finitely many points removed, which is path connected. Thus there exists a path in $L\cap C$ connecting $z$ and $w$.