{"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/automatic-bounds-on-constant-term-sequences","title":"Automatic Bounds on Constant Term Sequences Modulo Primes","arxiv_id":"2504.19031","date":"2025-04-26","proceeding":null,"authors":["Justin Offutt"],"abstract":"This paper provides counterexamples to a previously conjectured upper bound on the first index $n_0$ at which a zero appears in constant term sequences of the form $A_p(n) = ct(P^n) \\mod p$, where $P(t) \\in \\mathbb{Z}[t, t^{-1}]$. The conjecture posited that the first zero must occur at some index $n_0 < p^{\\text{deg}(P)}$. We prove an automaton state-based bound for univariate polynomials $n_0 < p^{\\kappa(P, p)}$, where $\\kappa(P, p)$ is the automaticity of $(A_p(n))_{n \\geq 0}$ over $\\mathbb{F}_p$. We support our theoretical results with randomized experiments on low degree Laurent polynomials and propose the $\\kappa(P, p)$ based bound as a practical alternative to the general worst case bound arising from the Rowland Zeilberger construction.","url_abs":"https://arxiv.org/abs/2504.19031v1","url_pdf":"https://arxiv.org/pdf/2504.19031v1.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":"automatic-bounds-on-constant-term-sequences","repo_url":"https://github.com/jcbopit/CTSExperiments","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}