{"paper":{"title":"Topology and arithmetic of resultants, II: the resultant $=1$ hypersurface (with an appendix by C. Cazanave)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.AT","math.GT","math.NT"],"primary_cat":"math.AG","authors_text":"Benson Farb, Jesse Wolfson","submitted_at":"2015-07-05T22:02:54Z","abstract_excerpt":"We consider the moduli space $\\mathcal{R}_n$ of pairs of monic, degree $n$ polynomials whose resultant equals $1$. We relate the topology of these algebraic varieties to their geometry and arithmetic. In particular, we compute their \\'{e}tale cohomology, the associated eigenvalues of Frobenius, and the cardinality of their set of $\\mathbb{F}_q$-points. When $q$ and $n$ are coprime, we show that the \\'etale cohomology of $\\mathcal{R}_{n/\\bar{\\mathbb{F}}_q}$ is pure, and of Tate type if and only if $q\\equiv 1$ mod $n$. We also deduce the values of these invariants for the finite field counterpar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.01283","kind":"arxiv","version":2},"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"}