{"paper":{"title":"On the Bateman-Horn conjecture for polynomials over large finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexei Entin","submitted_at":"2014-09-02T19:52:32Z","abstract_excerpt":"We prove an analogue of the classical Bateman-Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable $x$) polynomials $F_1,\\ldots,F_m\\in\\mathbf{F}_q[t][x]$, with $q$ odd, we show that the number of $f\\in\\mathbf{F}_q[t]$ of degree $n\\ge\\max(3,\\mathrm{deg}_t F_1,\\ldots,\\mathrm{deg}_t F_m)$ such that all $F_i(t,f)\\in\\mathbf{F}_q[t],1\\le i\\le m$ are irreducible is $$\\left(\\prod_{i=1}^m\\frac{\\mu_i}{N_i}\\right) q^{n+1}\\left(1+O_{m,\\,\\max\\mathrm{deg} F_i,\\,n}\\left(q^{-1/2}\\rig"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0846","kind":"arxiv","version":3},"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"}