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Hilbert’s Irreducibility Theorem via Random Walks
International Mathematics Research Notices ( IF 0.9 ) Pub Date : 2022-07-22 , DOI: 10.1093/imrn/rnac188
Lior Bary-Soroker 1 , Daniele Garzoni 1
Affiliation  

Let $G$ be a connected linear algebraic group over a number field $K$, let $\Gamma $ 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 $G/\textrm {R}_{u}(G)$ is either trivial or semisimple and $Z$ satisfies certain necessary conditions, then a long random walk on a Cayley graph of $\Gamma $ 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.

中文翻译:

通过随机游走的希尔伯特不可约定理

令 $G$ 为数域 $K$ 上的连通线性代数群,令 $\Gamma $ 为 $G(K)$ 的有限生成 Zariski 稠密子群,令 $Z\subseteq G(K)$ 为一个薄集,在塞尔的意义上。我们证明,如果 $G/\textrm {R}_{u}(G)$ 是平凡的或半简单的,并且 $Z$ 满足某些必要条件,那么在 $\Gamma $ 的 Cayley 图上的长随机游走命中$Z$ 的元素概率可忽略不计。我们推导出伽罗瓦覆盖、特征多项式和群动作中的不动点的推论。我们还在 $K$ 是全局函数字段的情况下证明了类似的结果。
更新日期:2022-07-22
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