Generic monodromy group of Riemann surfaces for inverses to entire functions of finite order

Abstract

We consider the vector space $E_{\rho,p}$ of entire functions of finite order, whose types are not more than $p>0$, endowed with Frechet topology, which is generated by a sequence of weighted norms. We call a function $f\in E_{\rho,p}$ {\it typical} if it is surjective and has an infinite number critical points such that each of them is non-degenerate and all the values of $f$ at these points are pairwise different. We prove that the set of all typical functions contains a set which is $G_\delta$ and dense in $E_{\rho,p}$. Furthermore, we show that inverse to any typical function has Riemann surface whose monodromy group coincides with finitary symmetric group of permutations of naturals, which is unsolvable in the following strong sense: it does not have a normal tower of subgroups, whose factor groups are or abelian or finite. As a consequence from these facts and Topological Galois Theory, we obtain that generically (in the above sense) for $f\in E_{\rho,p}$ the solution of equation $f(w)=z$ cannot be represented via $z$ and complex constants by a finite number of the following actions: algebraic operations (i.e., rational ones and solutions of polynomial equations) and quadratures (in particular, superpositions with elementary functions).