Let $U$ be
An answer to the open set of ${\bf C}^n$ defined by $P\neq 0$. Take two points $a$ and $b\in U$more general question could be in three steps.To show that
1) If $U$ X$is a complex algebraic variety, connected for the Zariski topology, then it suffices to show that the intersection$L\cap U$is connected , where$L$is the line$(ab)$. Write a point$p$of for the line as$a+tu$, where$u=b-a$usual topology, an important and$t\in{\bf C}$. This describes$L\cap U$as the complement nontrivial fact that can be found in${\bf C}$of the locus defined by the equation$P(a+tu)=0$; the latter is finiteMumford's Red Book, or in Shafarevich's one. So it remains to show that the 2) The complement to a finite set of point closed Zariski subset in the plane an irreducible variety is connected . This is almost obvious: try to go straight away from$a$to$b$and when you hit one of for the removed points, make a small detour (a half circleZariski topology. 3) The varieties$SL(n,{\bf C})$and resume to your straight path.$SL(N,{\bf C})^N\$ are irreducible