{"paper":{"title":"On algebraic curves A(x)-B(y)=0 of genus zero","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CV","math.DS"],"primary_cat":"math.NT","authors_text":"Fedor Pakovich","submitted_at":"2015-05-05T13:47:35Z","abstract_excerpt":"Using a geometric approach involving Riemann surface orbifolds, we provide lower bounds for the genus of an irreducible algebraic curve of the form $E_{A,B}:\\, A(x)-B(y)=0$, where $A, B\\in\\mathbb C(z)$. We also investigate \"series\" of curves $E_{A,B}$ of genus zero, where by a series we mean a family with the \"same\" $A$. We show that for a given rational function $A$ a sequence of rational functions $B_i$, such that ${\\rm deg}\\, B_i \\rightarrow \\infty$ and all the curves $A(x)-B_i(y)=0$ are irreducible and have genus zero, exists if and only if the Galois closure of the field extension $\\mathb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01007","kind":"arxiv","version":5},"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"}