{"paper":{"title":"Integral polynomials with small discriminants and resultants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Friedrich G\\\"otze, Vasili Bernik, Victor Beresnevich","submitted_at":"2015-01-23T11:05:33Z","abstract_excerpt":"Let $n\\in\\mathbb{N}$ be fixed, $Q>1$ be a real parameter and $\\mathcal{P}_n(Q)$ denote the set of polynomials over $\\mathbb{Z}$ of degree $n$ and height at most $Q$. In this paper we investigate the following counting problems regarding polynomials with small discriminant $D(P)$ and pairs of polynomials with small resultant $R(P_1,P_2)$:\n  (i) given $0\\le v\\le n-1$ and a sufficiently large $Q$, estimate the number of polynomials $P\\in\\mathcal{P}_n(Q)$ such that $$0<|D(P)|\\le Q^{2n-2-2v};$$ (ii) given $0\\le w\\le n$ and a sufficiently large $Q$, estimate the number of pairs of polynomials $P_1,P"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.05767","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"}