Hilbert's 13th Problem for Algebraic Groups

TL;DR

This paper extends the concept of resolvent degree from symmetric groups to all algebraic groups over arbitrary fields, establishing an upper bound of 5 for connected groups and exploring its dependence on the base field.

Contribution

It generalizes the resolvent degree to algebraic groups over any field and proves an upper bound of 5 for connected groups, opening questions for non-connected groups.

Findings

entity of resolvent degree for algebraic groups over arbitrary fields.

Upper bound of 5 for connected algebraic groups.

Open problem on whether resolvent degree exceeds 1 for any algebraic group.

Abstract

The algebraic form of Hilbert's 13th Problem asks for the resolvent degree $\text{rd}(n)$ of the general polynomial $f(x) = x^n + a_1 x^{n-1} + \ldots + a_n$ of degree $n$, where $a_1, \ldots, a_n$ 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 $\mathbb C(a_1, \ldots, a_n)$ and adjoining algebraic functions in $\leq d$ variables at each step. Recently Farb and Wolfson defined the resolvent degree $\text{rd}_k(G)$ of any finite group $G$ and any base field $k$ of characteristic $0$. In this setting $\text{rd}(n) = \text{rd}_{\mathbb C}(S_n)$, where $S_n$ denotes the symmetric group. In this paper we define $\text{rd}_k(G)$ for every algebraic group $G$ over an arbitrary field $k$, investigate the dependency of this quantity on $k$ and show that $\text{rd}_k(G) \leq 5$ for any field $k$ and any connected group $G$. The question of whether $\text{rd}_k(G)$ can be bigger than $1$ for any field $k$ and any algebraic group $G$ over $k$ (not necessarily connected) remains open.