{"paper":{"title":"On the Enumeration of Irreducible Polynomials over $\\text{GF}(q)$ with Prescribed Coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Robert Granger","submitted_at":"2016-10-21T17:50:03Z","abstract_excerpt":"We present an efficient deterministic algorithm which outputs exact expressions in terms of $n$ for the number of monic degree $n$ irreducible polynomials over $\\mathbb{F}_{q}$ of characteristic $p$ for which the first $l < p$ coefficients are prescribed, provided that $n$ is coprime to $p$. Each of these counts is $\\frac{1}{n}(q^{n-l} + \\mathcal{O}(q^{n/2}))$. The main idea behind the algorithm is to associate to an equivalent problem a set of Artin-Schreier curves defined over $\\mathbb{F}_q$ whose number of $\\mathbb{F}_{q^n}$-rational affine points must be combined. This is accomplished by c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.06878","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"}