{"paper":{"title":"Estimating the Number Of Roots of Trinomials over Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Sean Owen, Zander Kelley","submitted_at":"2015-10-06T20:47:38Z","abstract_excerpt":"We show that univariate trinomials $x^n + ax^s + b \\in \\mathbb{F}_q[x]$ can have at most $\\delta \\Big\\lfloor \\frac{1}{2} +\\sqrt{\\frac{q-1}{\\delta}} \\Big\\rfloor$ distinct roots in $\\mathbb{F}_q$, where $\\delta = \\gcd(n, s, q - 1)$. We also derive explicit trinomials having $\\sqrt{q}$ roots in $\\mathbb{F}_q$ when $q$ is square and $\\delta=1$, thus showing that our bound is tight for an infinite family of finite fields and trinomials. Furthermore, we present the results of a large-scale computation which suggest that an $O(\\delta \\log q)$ upper bound may be possible for the special case where $q$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.01758","kind":"arxiv","version":4},"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"}