{"paper":{"title":"The emergence of 4-cycles in polynomial maps over the extended integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrew Best, Benjamin L. Weiss, Jasmine Powell, Patrick Dynes, Steven J. Miller","submitted_at":"2015-07-13T20:01:26Z","abstract_excerpt":"Let $f(x) \\in \\mathbb{Z}[x]$; for each integer $\\alpha$ it is interesting to consider the number of iterates $n_{\\alpha}$, if possible, needed to satisfy $f^{n_{\\alpha}}(\\alpha) = \\alpha$. The sets $\\{\\alpha, f(\\alpha), \\ldots, f^{n_{\\alpha} - 1}(\\alpha), \\alpha\\}$ generated by the iterates of $f$ are called cycles. For $\\mathbb{Z}[x]$ it is known that cycles of length 1 and 2 occur, and no others. While much is known for extensions to number fields, we concentrate on extending $\\mathbb{Z}$ by adjoining reciprocals of primes. Let $\\mathbb{Z}[1/p_1, \\ldots, 1/p_n]$ denote $\\mathbb{Z}$ extended "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03597","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"}