论文标题
希尔伯特通过随机步行的不可约性定理
Hilbert's irreducibility theorem via random walks
论文作者
论文摘要
让$ g $为一个数字字段$ k $的连接线性代数群，让$γ$成为有限生成的zariski密度$ g（k）$，而让$ z \ subseteq g（k）$是薄的集合。我们证明，如果$ g/\ mathrm {r} _u（g）$是半imple，$ z $满足某些必要条件，则在$γ$的cayley图上进行长时间随机步行，键击中了$ z $的元素，$ z $的概率可忽略不计。我们在小组动作中将推论推论为Galois覆盖，特征多项式和固定点。在$ k $是全球函数字段的情况下，我们还证明了类似的结果。
Let $G$ be a connected linear algebraic group over a number field $K$, let $Γ$ be a finitely generated Zariski dense subgroup of $G(K)$ and let $Z\subseteq G(K)$ be a thin set, in the sense of Serre. We prove that, if $G/\mathrm{R}_u(G)$ is semisimple and $Z$ satisfies certain necessary conditions, then a long random walk on a Cayley graph of $Γ$ hits elements of $Z$ with negligible probability. We deduce corollaries to Galois covers, characteristic polynomials, and fixed points in group actions. We also prove analogous results in the case where $K$ is a global function field.
