{"paper":{"title":"Rational values of transcendental functions and arithmetic dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Gareth Boxall, Gareth Jones, Harry Schmidt","submitted_at":"2018-08-23T09:42:42Z","abstract_excerpt":"We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work with p-adic methods to obtain a lower bound of the form $cD^{n/4 - \\varepsilon}$ on the degree of the splitting field of $P^{\\circ n}(z)=P^{\\circ n}(\\alpha)$, where $P$ is a polynomial of degree $D\\geq 2$ over a number field, $P^{\\circ n}$ is its $n$-th iterate and $c$ depends effectively on $P, \\alpha$ and $\\varepsilon$. Our $c$ is positive for each algebr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.07676","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"}