论文标题
希尔伯特（Hilbert）对于代数群体的第13个问题
Hilbert's 13th Problem for Algebraic Groups
论文作者
论文摘要
希尔伯特（Hilbert）第13个问题的代数形式要求将一般多项式$ f（x）= x^n + a_1 x^{n + a_1 x^{n-1} + \ ldots + a_n $ a_1 $ n $ n $ a_1 $ a_1，\ a_1，\ ldots，a__n $ a_n $ a_n $变量。解决方案是最小整数$ d $，因此，$ f（x）$的每个根都可以以有限数量的步骤获得，从$ \ mathbb c（a_1，\ ldots，a_n）$开始，并在每个步骤中以$ \ leq d $ variables中的代数函数相邻。最近，Farb和Wolfson定义了任何有限组$ g $的分辨率$ \ text {rd} _k（g）$和任何特征性$ 0 $的基本场$ k $。在此设置中，$ \ text {rd}（n）= \ text {rd} _ {\ mathbb c}（s_n）$，其中$ s_n $表示对称组。在本文中，我们为任意字段$ k $上的每个代数组$ g $定义$ \ text {rd} _k（g）$，调查此数量对$ k $的依赖性，并显示$ \ text {rd} _k（g）\ leq 5 $对于任何字段$ k $和任何连接的组$ g $ $ g $。对于任何字段$ k $而言，$ \ text {rd} _k（g）$是否大于$ 1 $，任何代数组$ g $ over $ k $（不一定连接）仍然开放。
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.
