{"paper":{"title":"On the factorization of iterates of $x^d+c$ in large degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Wade Hindes","submitted_at":"2025-08-07T19:08:56Z","abstract_excerpt":"Let $K$ be a function field of a curve in characteristic zero or a number field over which the $abc$-conjecture holds, fix $\\alpha\\in K$, and let $f_{d,c}(x)=x^d+c$ for some $d\\geq2$ and some $c\\in K$. Then for many $c$ and $d$, we prove that $f_{d,c}^n(x)-\\alpha$ has at most $d$ factors in $K[x]$ for all $n\\geq1$. For example, when $\\alpha=0$ we prove that the set \\[\\Big\\{d\\,:\\, f_{d,c}^n(x)\\;\\text{has at most $d$ factors in $K[x]$ for all $n\\geq1$ and all $h(c)>0$}\\Big\\}\\] has positive asymptotic density. We then apply this result to compute the density of prime divisors in certain forward o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.05795","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.05795/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}