Hilbert's 13th problem for algebraic groups

Abstract

The algebraic form of Hilbert's 13th problem asks for the resolvent degree $\mathrm{RD}(n)$ of the general polynomial $f(x)=x^n+a_1{x}^{n-1}+\cdots +a_n$ of degree $n$, where $a_1,\dots ,a_n$ are independent variables. The resolvent degree is the minimal integer $d$ such that every root of $f(x)$ can be obtained in a finite number of steps, starting with $\mathbb{C}(a_1,\dots ,a_n)$ and adjoining algebraic functions in $⩽d$ variables at each step. Recently Farb and Wolfson defined the resolvent degree ${\mathrm{RD}}_k(G)$ for every finite group $G$ and any base field $k$ of characteristic $0$. In this setting $\mathrm{RD}(n)={\mathrm{RD}}_{\mathbb{C}}(S_n)$, where $S_n$ denotes the symmetric group. In this paper we extend their definition of ${\mathrm{RD}}_k(G)$ to an arbitrary algebraic {group} $G$ over an arbitrary field $k$. We investigate the dependency of this quantity on $k$ and show that ${\mathrm{RD}}_k(G)⩽5$ for any field $k$ and any connected group $G$. The question whether ${\mathrm{RD}}_k(G)$ can be bigger than $1$ for any field $k$ and any algebraic group $G$ over $k$ (not necessarily connected) remains open.