{"paper":{"title":"On Hilbert's irreducibility theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Abel Castillo, Rainer Dietmann","submitted_at":"2016-01-31T20:26:06Z","abstract_excerpt":"In this paper we obtain new quantitative forms of Hilbert's Irreducibility Theorem. In particular, we show that if $f(X, T_1, \\ldots, T_s)$ is an irreducible polynomial with integer coefficients, having Galois group $G$ over the function field $\\mathbb{Q}(T_1, \\ldots, T_s)$, and $K$ is any subgroup of $G$, then there are at most $O_{f, \\varepsilon}(H^{s-1+|G/K|^{-1}+\\varepsilon})$ specialisations $\\mathbf{t} \\in \\mathbb{Z}^s$ with $|\\mathbf{t}| \\le H$ such that the resulting polynomial $f(X)$ has Galois group $K$ over the rationals."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00314","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}