{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-local-global-principle-in-the-dynamics-of","title":"A local-global principle in the dynamics of quadratic polynomials","arxiv_id":"1508.03830","date":"2015-08-16","proceeding":null,"authors":["David Krumm"],"abstract":"Let $K$ be a number field, $f\\in K[x]$ a quadratic polynomial, and $n\\in\\{1,2,3\\}$. We show that if $f$ has a point of period $n$ in every non-archimedean completion of $K$, then $f$ has a point of period $n$ in $K$. For $n\\in\\{4,5\\}$ we show that there exist at most finitely many linear conjugacy classes of quadratic polynomials over $K$ for which this local-global principle fails. By considering a stronger form of this principle, we strengthen global results obtained by Morton and Flynn-Poonen-Schaefer in the case $K=\\mathbf Q$. More precisely, we show that for every quadratic polynomial $f\\in\\mathbf Q[x]$ there exist infinitely many primes $p$ such that $f$ does not have a point of period 4 in the $p$-adic field $\\mathbf Q_p$. Conditional on knowing all rational points on a particular curve of genus 11, the same result is proved for points of period 5.","url_abs":"http://arxiv.org/abs/1508.03830v2","url_pdf":"http://arxiv.org/pdf/1508.03830v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-local-global-principle-in-the-dynamics-of","repo_url":"https://github.com/davidkrumm/local-global","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}