论文标题
在专业化的对称Galois组
Symmetric Galois Groups Under Specialization
论文作者
论文摘要
给定一个不可约束的双变量多项式$ f（t，x）\ in \ mathbb {q} [t，x] $，$ f（t_0，x）$的Galois组出现了哪些组$ h $，用于无限的许多$ t_0 \ in \ mathbb {q} $？以上作为$ f（t_0，x）$，$ t_0 \ in \ mathbb {q} $的Galois组出现的$ H $多久出现？我们给出了$ f $ $ x $ -Degree的答案，其交替或对称的galois组超过$ \ mathbb {q}（t）$。这是通过确定覆盖物的低属子覆盖物$ \ tilde {x} \ rightarrow \ mathbb {p}^1 _ {\ mathbb {c}} $，并使用交替或对称单型组组。
Given an irreducible bivariate polynomial $f(t,x)\in \mathbb{Q}[t,x]$, what groups $H$ appear as the Galois group of $f(t_0,x)$ for infinitely many $t_0\in \mathbb{Q}$? How often does a group $H$ as above appear as the Galois group of $f(t_0,x)$, $t_0\in \mathbb{Q}$? We give an answer for $f$ of large $x$-degree with alternating or symmetric Galois group over $\mathbb{Q}(t)$. This is done by determining the low genus subcovers of coverings $\tilde{X}\rightarrow \mathbb{P}^1_{\mathbb{C}}$ with alternating or symmetric monodromy groups.
