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

TL;DR

This paper studies the monodromy groups of inverse functions of typical entire functions of finite order, showing that generically these groups are highly complex and unsolvable, impacting the solvability of related equations.

Contribution

It proves that the set of typical functions with certain properties is dense and G_delta, and that their inverse functions have monodromy groups isomorphic to the finitary symmetric group, which is strongly unsolvable.

Findings

Typical functions form a dense G_delta set in the function space.

Inverse functions of typical functions have monodromy groups isomorphic to the finitary symmetric group.

Solutions to equations involving these functions cannot be expressed via algebraic operations and quadratures.

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).