{"paper":{"title":"A variation on Magnus' theorem and its generalizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Vered Moskowicz","submitted_at":"2018-10-18T17:56:02Z","abstract_excerpt":"Let $k$ be a field of characteristic zero, and let $f: k[x,y] \\to k[x,y]$, $f: (x,y) \\mapsto (p,q)$, be a $k$-algebra endomorphism having an invertible Jacobian. Write $p=a_ny^n+\\cdots+a_1y+a_0$, where $n=deg_y(p) \\in \\mathbb{N}$, $a_i \\in k[x]$, $0 \\leq i \\leq n$, $a_n \\neq 0$, and $q=c_ry^r+\\cdots+c_1y+c_0$, where $r=deg_y(q) \\in \\mathbb{N}$, $c_i \\in k[x]$, $0 \\leq i \\leq r$, $c_r \\neq 0$. Denote the set of prime numbers by $P$. Under two mild conditions, we prove that, if $\\gcd(\\gcd(n,deg_x(a_n)),\\gcd(r,deg_x(c_r))) \\in \\{1,8\\} \\cup P \\cup 2P$, then $f$ is an automorphism of $k[x,y]$. Remo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.08202","kind":"arxiv","version":2},"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"}