Algebraic functions, configuration spaces, Teichmüller spaces, and new holomorphically combinatorial invariants

It is proved that, for $n\ge 4$, the function $u=u_n(z)$, $z=(z_1,\dots,z_n)\in{\mathbb{C}}^n$, defined by the equation $u^n +z_1 u^{n-1} +\dots + z_n=0$ cannot be a branch of an entire algebraic function $g$ on ${\mathbb{C}}^n$ that is a composition of entire algebraic functions depending on fewer than $n-1$ variables and has the same discriminant set as $u_n$. A key role is played by a description of holomorphic maps between configuration spaces of ${\mathbb{C}}$ and ${\mathbb{CP}}^1$, which, in turn, involves Teichmüller spaces and new holomorphically combinatorial invariants of complex spaces.