{"paper":{"title":"Permutation Polynomials of $\\Bbb F_{q^2}$ of the form $a{\\tt X}+{\\tt X}^{r(q-1)+1}$","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":"2015-10-01T22:10:54Z","abstract_excerpt":"Let $q$ be a prime power, $2\\le r\\le q$, and $f=a{\\tt X}+{\\tt X}^{r(q-1)+1}\\in\\Bbb F_{q^2}[{\\tt X}]$, where $a\\ne 0$. The conditions on $r,q,a$ that are necessary and sufficient for $f$ to be a permutation polynomial (PP) of ${\\Bbb F}_{q^2}$ are not known. (Such conditions are known under an additional assumption that $a^{q+1}=1$.) In this paper, we prove the following: (i) If $f$ is a PP of ${\\Bbb F}_{q^2}$, then $\\text{gcd}(r,q+1)>1$ and $(-a)^{(q+1)/\\text{gcd}(r,q+1)}\\ne 1$. (ii) For a fixed $r>2$ and subject to the conditions that $q+1\\equiv 0\\pmod r$ and $a^{q+1}\\ne 1$, there are only fin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00437","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"}