[{"month":"10","title":"Common valuations of division polynomials","ddc":["510"],"acknowledgement":"Silverman, and Paul Voutier for the comments on the earlier version of this paper. The first author acknowledges the support by Dioscuri programme initiated by the Max Planck Society, jointly managed with the National Science Centre (Poland), and mutually funded by the Polish Ministry of Science and Higher Education and the German Federal Ministry of Education and Research. The second author has been supported by MIUR (Italy) through PRIN 2017 ‘Geometric, algebraic and analytic methods in arithmetic’ and has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413.","department":[{"_id":"TiBr"}],"date_published":"2025-10-01T00:00:00Z","article_processing_charge":"Yes (via OA deal)","year":"2025","file_date_updated":"2025-12-30T06:45:47Z","article_type":"original","corr_author":"1","file":[{"date_created":"2025-12-30T06:45:47Z","access_level":"open_access","content_type":"application/pdf","date_updated":"2025-12-30T06:45:47Z","file_name":"2025_ProceedingsRoyalSocEdinburghA_Naskrecki.pdf","relation":"main_file","file_size":477624,"file_id":"20878","success":1,"creator":"dernst","checksum":"c5ec6e29aca2fb4533cb95fac409a0b2"}],"publication_identifier":{"issn":["0308-2105"],"eissn":["1473-7124"]},"publication_status":"published","date_created":"2023-01-16T11:45:22Z","OA_type":"hybrid","intvolume":"       155","has_accepted_license":"1","issue":"5","scopus_import":"1","project":[{"call_identifier":"H2020","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","name":"IST-BRIDGE: International postdoctoral program"}],"publication":"Proceedings of the Royal Society of Edinburgh Section A: Mathematics","quality_controlled":"1","researchdata_availability":"no","PlanS_conform":"1","type":"journal_article","isi":1,"citation":{"chicago":"Naskręcki, Bartosz, and Matteo Verzobio. “Common Valuations of Division Polynomials.” <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>. Cambridge University Press, 2025. <a href=\"https://doi.org/10.1017/prm.2024.7\">https://doi.org/10.1017/prm.2024.7</a>.","short":"B. Naskręcki, M. Verzobio, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 155 (2025) 1646–1660.","mla":"Naskręcki, Bartosz, and Matteo Verzobio. “Common Valuations of Division Polynomials.” <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>, vol. 155, no. 5, Cambridge University Press, 2025, pp. 1646–60, doi:<a href=\"https://doi.org/10.1017/prm.2024.7\">10.1017/prm.2024.7</a>.","ista":"Naskręcki B, Verzobio M. 2025. Common valuations of division polynomials. Proceedings of the Royal Society of Edinburgh Section A: Mathematics. 155(5), 1646–1660.","ieee":"B. Naskręcki and M. Verzobio, “Common valuations of division polynomials,” <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>, vol. 155, no. 5. Cambridge University Press, pp. 1646–1660, 2025.","ama":"Naskręcki B, Verzobio M. Common valuations of division polynomials. <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>. 2025;155(5):1646-1660. doi:<a href=\"https://doi.org/10.1017/prm.2024.7\">10.1017/prm.2024.7</a>","apa":"Naskręcki, B., &#38; Verzobio, M. (2025). Common valuations of division polynomials. <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/prm.2024.7\">https://doi.org/10.1017/prm.2024.7</a>"},"status":"public","doi":"10.1017/prm.2024.7","ec_funded":1,"OA_place":"publisher","keyword":["Elliptic curves","Néron models","division polynomials","height functions","discrete valuation rings"],"abstract":[{"lang":"eng","text":"In this note, we prove a formula for the cancellation exponent  kv,n between division polynomials  ψn  and  ϕn  associated with a sequence  {nP}n∈N of points on an elliptic curve  E  defined over a discrete valuation field  K. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields."}],"language":[{"iso":"eng"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","image":"/images/cc_by.png"},"_id":"12311","author":[{"first_name":"Bartosz","full_name":"Naskręcki, Bartosz","last_name":"Naskręcki"},{"first_name":"Matteo","full_name":"Verzobio, Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","orcid":"0000-0002-0854-0306","last_name":"Verzobio"}],"arxiv":1,"supplementarymaterial":"no","date_updated":"2026-07-16T08:43:40Z","fulldoi":"https://doi.org/10.1017/prm.2024.7","publisher":"Cambridge University Press","das_tickbox":"0","day":"01","oa_version":"Published Version","volume":155,"external_id":{"isi":["001174907100001"],"arxiv":["2203.02015"]},"oa":1,"page":"1646-1660"},{"doi":"10.1016/j.jnt.2022.04.006","keyword":["Pseudo-polynomials","Chinese remainder theorem","Ruzsa’s conjecture"],"OA_place":"repository","language":[{"iso":"eng"}],"abstract":[{"text":"We explore two questions about pseudo-polynomials, which\r\nare functions f : N → Z such that k divides f(n + k) −\r\nf(n) for all n, k. First, for certain arbitrarily sparse sets R, we\r\nconstruct pseudo-polynomials f with p|f(n) for some n only if\r\np ∈ R. This implies that not all pseudo-polynomials satisfy an\r\nassumption of a recent paper of Kowalski and Soundararajan.\r\nWe also consider α-primary pseudo-polynomials, where the\r\npseudo-polynomial condition is only required for k lying in\r\na set of primes of density α. We show that if an α-primary\r\npseudo-polynomial is O(e(β−)n), where β = √7\r\n3 − 1\r\n6 ≈ 0.715,\r\nthen it is a polynomial.","lang":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve","first_name":"Vivian Zieve","last_name":"Kuperberg"}],"_id":"22196","arxiv":1,"date_updated":"2026-07-14T11:08:14Z","publisher":"Elsevier","fulldoi":"https://doi.org/10.1016/j.jnt.2022.04.006","day":"18","oa_version":"Preprint","volume":241,"external_id":{"arxiv":["2006.02527"]},"page":"531-541","oa":1,"month":"05","title":"On pseudo-polynomials divisible only by a sparse set of primes and α-primary pseudo-polynomials","date_published":"2022-05-18T00:00:00Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2006.02527","open_access":"1"}],"article_processing_charge":"No","article_type":"original","year":"2022","date_created":"2026-06-29T12:58:07Z","publication_status":"published","publication_identifier":{"issn":["0022-314X"]},"OA_type":"green","extern":"1","intvolume":"       241","quality_controlled":"1","publication":"Journal of Number Theory","scopus_import":"1","type":"journal_article","citation":{"short":"V.Z. Kuperberg, Journal of Number Theory 241 (2022) 531–541.","chicago":"Kuperberg, Vivian Zieve. “On Pseudo-Polynomials Divisible Only by a Sparse Set of Primes and α-Primary Pseudo-Polynomials.” <i>Journal of Number Theory</i>. Elsevier, 2022. <a href=\"https://doi.org/10.1016/j.jnt.2022.04.006\">https://doi.org/10.1016/j.jnt.2022.04.006</a>.","mla":"Kuperberg, Vivian Zieve. “On Pseudo-Polynomials Divisible Only by a Sparse Set of Primes and α-Primary Pseudo-Polynomials.” <i>Journal of Number Theory</i>, vol. 241, Elsevier, 2022, pp. 531–41, doi:<a href=\"https://doi.org/10.1016/j.jnt.2022.04.006\">10.1016/j.jnt.2022.04.006</a>.","ista":"Kuperberg VZ. 2022. On pseudo-polynomials divisible only by a sparse set of primes and α-primary pseudo-polynomials. Journal of Number Theory. 241, 531–541.","ieee":"V. Z. Kuperberg, “On pseudo-polynomials divisible only by a sparse set of primes and α-primary pseudo-polynomials,” <i>Journal of Number Theory</i>, vol. 241. Elsevier, pp. 531–541, 2022.","ama":"Kuperberg VZ. On pseudo-polynomials divisible only by a sparse set of primes and α-primary pseudo-polynomials. <i>Journal of Number Theory</i>. 2022;241:531-541. doi:<a href=\"https://doi.org/10.1016/j.jnt.2022.04.006\">10.1016/j.jnt.2022.04.006</a>","apa":"Kuperberg, V. Z. (2022). On pseudo-polynomials divisible only by a sparse set of primes and α-primary pseudo-polynomials. <i>Journal of Number Theory</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jnt.2022.04.006\">https://doi.org/10.1016/j.jnt.2022.04.006</a>"},"status":"public"}]