{"paper":{"title":"Scarcity of finite orbits for rational functions over a number field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"J.K. Canci, Sebastian Troncoso, Solomon Vishkautsan","submitted_at":"2017-11-09T22:28:51Z","abstract_excerpt":"Let $\\phi$ be a an endomorphism of degree $d\\geq{2}$ of the projective line, defined over a number field $K$. Let $S$ be a finite set of places of $K$, including the archimedean places, such that $\\phi$ has good reduction outside of $S$. The article presents two main results: the first result is a bound on the number of $K$-rational preperiodic points of $\\phi$ in terms of the cardinality of the set $S$ and the degree $d$ of the endomorphism $\\phi$. This bound is quadratic in terms of $d$ which is a significant improvement to all previous bounds on the number of preperiodic points in terms of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.04649","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"}