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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 $F$."},"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":"1208.6108","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-08-30T08:25:01Z","cross_cats_sorted":[],"title_canon_sha256":"1d5ab625f94b0d858042d9f2fa1565eadc6ea89e9f754b002bbf52a66c9335ab","abstract_canon_sha256":"a79a53d8bc6fe1a91f757bc50dcd756d91ad53418ef73bae04145816a2c2207d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:47.596277Z","signature_b64":"J4DK8B837KBz+wRjGAHwIgqkKqlh29wg065iX/QH8JFZ9eKuAzBn3CUlM9j+UY+4YMf+9By2NPH39NOtgqpABQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e44cbd28cbdf1ad18d12a4e6fb37174550dc92573ab9ffb1f23fc5dd3160f3cf","last_reissued_at":"2026-05-18T03:08:47.595826Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:47.595826Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\mathbb{C}$. 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