{"paper":{"title":"Algebraic curves $A^{\\circ l}(x)-U(y)=0$ and arithmetic of orbits of rational functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CV","math.NT"],"primary_cat":"math.DS","authors_text":"Fedor Pakovich","submitted_at":"2018-01-06T08:05:43Z","abstract_excerpt":"We give a description of pairs of complex rational functions $A$ and $U$ of degree at least two such that for every $d\\geq 1$ the algebraic curve $A^{\\circ d}(x)-U(y)=0$ has a factor of genus zero or one. In particular, we show that if $A$ is not a `generalized Latt\\`es map', then this condition is satisfied if and only if there exists a rational function $V$ such that $U\\circ V=A^{\\circ l}$ for some $l\\geq 1.$ We also prove a version of the dynamical Mordell-Lang conjecture, concerning intersections of orbits of points from $\\mathbb P^1(K)$ under iterates of $A$ with the value set $U(\\mathbb "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.01985","kind":"arxiv","version":2},"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"}