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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1810.08202","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-10-18T17:56:02Z","cross_cats_sorted":[],"title_canon_sha256":"0912280fe9562cb139d1730d80269289cdbc31f8bf0ddeb9c4bf7a2890c89dfe","abstract_canon_sha256":"86f8b3a5c2525bd2385f6c95aca3caefe93bc665483f2f57445e418820d65712"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:24.548462Z","signature_b64":"WhqCv3d17wYW3vxiVAu/L5m5oAmwIET/wl+b5fluwlH3AvTPeZbvs8FlcH1x/o2MjET3hLMSDa573Rc3eK29Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04b5f0f8ff296d65700b8e7d05ab67e2f2f9656ed7482670d70588c33f127e8b","last_reissued_at":"2026-05-18T00:02:24.547607Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:24.547607Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $k[x,y]$. 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