{"paper":{"title":"Picard theorems for Keller mappings in dimension two and the phantom curve","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Ronen Peretz","submitted_at":"2012-08-30T08:25:01Z","abstract_excerpt":"Let $F=(P,Q)\\in\\mathbb{C}[X,Y]^{2}$ be a polynomial mapping over the complex field $\\mathbb{C}$. Suppose that $$ \\det\\,J_{F}(X,Y):=\\frac{\\partial P}{\\partial X}\\frac{\\partial Q}{\\partial Y}- \\frac{\\partial P}{\\partial Y}\\frac{\\partial Q}{\\partial X}=a\\in\\mathbb{C}^{\\times}. $$ A mapping that satisfies the assumptions above is called a Keller mapping. In this paper we estimate the size of the co-image of $F$. We give a sufficient condition for surjectivity of Keller mappings in terms of its Phantom curve. This curve is closely related to the asymptotic variety of $F$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.6108","kind":"arxiv","version":3},"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"}