{"paper":{"title":"Determination of a Type of Permutation Trinomials over Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Xiang-dong Hou","submitted_at":"2013-09-13T18:15:14Z","abstract_excerpt":"Let $f=a{\\tt x} +b{\\tt x}^q+{\\tt x}^{2q-1}\\in\\Bbb F_q[{\\tt x}]$. We find explicit conditions on $a$ and $b$ that are necessary and sufficient for $f$ to be a permutation polynomial of $\\Bbb F_{q^2}$. This result allows us to solve a related problem. Let $g_{n,q}\\in\\Bbb F_p[{\\tt x}]$ ($n\\ge 0$, $p=\\text{char}\\,\\Bbb F_q$) be the polynomial defined by the functional equation $\\sum_{c\\in\\Bbb F_q}({\\tt x}+c)^n=g_{n,q}({\\tt x}^q-{\\tt x})$. We determine all $n$ of the form $n=q^\\alpha-q^\\beta-1$, $\\alpha>\\beta\\ge 0$, for which $g_{n,q}$ is a permutation polynomial of $\\Bbb F_{q^2}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3530","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"}