{"paper":{"title":"Birational morphisms of the plane","license":"","headline":"","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"Jie-Tai Yu, Vladimir Shpilrain","submitted_at":"2003-09-07T01:04:46Z","abstract_excerpt":"Let A^2 be the affine plane over a field K of characteristic 0. Birational morphisms of A^2 are mappings A^2 \\to A^2 given by polynomial mappings \\phi of the polynomial algebra K[x,y] such that for the quotient fields, one has K(\\phi(x), \\phi(y)) = K(x,y).\n Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping \\tau_x given by x \\to x, y \\to xy. For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of \\tau_x. This question was answered in the negative by P. Russell (in an inform"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0309125","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"}