{"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-finiteness-theorem-for-specializations-of","title":"A finiteness theorem for specializations of dynatomic polynomials","arxiv_id":"1805.11152","date":"2018-05-28","proceeding":null,"authors":["David Krumm"],"abstract":"Let $t$ and $x$ be indeterminates, let $\\phi(x)=x^2+t\\in\\mathbb Q(t)[x]$, and for every positive integer $n$ let $\\Phi_n(t,x)$ denote the $n^{\\text{th}}$ dynatomic polynomial of $\\phi$. Let $G_n$ be the Galois group of $\\Phi_n$ over the function field $\\mathbb Q(t)$, and for $c\\in\\mathbb Q$ let $G_{n,c}$ be the Galois group of the specialized polynomial $\\Phi_n(c,x)$. It follows from Hilbert's irreducibility theorem that for fixed $n$ we have $G_n\\cong G_{n,c}$ for every $c$ outside a thin set $E_n\\subset\\mathbb Q$. By earlier work of Morton (for $n=3$) and the present author (for $n=4$), it is known that $E_n$ is infinite if $n\\le 4$. In contrast, we show here that $E_n$ is finite if $n\\in\\{5,6,7,9\\}$. As an application of this result we show that, for these values of $n$, the following holds with at most finitely many exceptions: for every $c\\in\\mathbb Q$, more than $81\\%$ of prime numbers $p$ have the property that the polynomial $x^2+c$ does not have a point of period $n$ in the $p$-adic field $\\mathbb Q_p$.","url_abs":"http://arxiv.org/abs/1805.11152v2","url_pdf":"http://arxiv.org/pdf/1805.11152v2.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-finiteness-theorem-for-specializations-of","repo_url":"https://github.com/davidkrumm/finiteness_dynatomic","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}