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arXiv · 2202.05010

Hilbert's irreducibility theorem via random walks

Lior Bary-Soroker·Daniele Garzoni
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Abstract

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 $G/\mathrm{R}_u(G)$ is 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.

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Lior Bary-Soroker, Daniele Garzoni. 2022-02-10. Hilbert's irreducibility theorem via random walks. https://arxiv.org/abs/2202.05010

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