---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '12311'
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.
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.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Bartosz
full_name: Naskręcki, Bartosz
last_name: Naskręcki
- first_name: Matteo
full_name: Verzobio, Matteo
id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
last_name: Verzobio
orcid: 0000-0002-0854-0306
citation:
ama: 'Naskręcki B, Verzobio M. Common valuations of division polynomials. Proceedings
of the Royal Society of Edinburgh Section A: Mathematics. 2025;155(5):1646-1660.
doi:10.1017/prm.2024.7'
apa: 'Naskręcki, B., & Verzobio, M. (2025). Common valuations of division polynomials.
Proceedings of the Royal Society of Edinburgh Section A: Mathematics. Cambridge
University Press. https://doi.org/10.1017/prm.2024.7'
chicago: 'Naskręcki, Bartosz, and Matteo Verzobio. “Common Valuations of Division
Polynomials.” Proceedings of the Royal Society of Edinburgh Section A: Mathematics.
Cambridge University Press, 2025. https://doi.org/10.1017/prm.2024.7.'
ieee: 'B. Naskręcki and M. Verzobio, “Common valuations of division polynomials,”
Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol.
155, no. 5. Cambridge University Press, pp. 1646–1660, 2025.'
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.'
mla: 'Naskręcki, Bartosz, and Matteo Verzobio. “Common Valuations of Division Polynomials.”
Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol.
155, no. 5, Cambridge University Press, 2025, pp. 1646–60, doi:10.1017/prm.2024.7.'
short: 'B. Naskręcki, M. Verzobio, Proceedings of the Royal Society of Edinburgh
Section A: Mathematics 155 (2025) 1646–1660.'
corr_author: '1'
das_tickbox: '0'
date_created: 2023-01-16T11:45:22Z
date_published: 2025-10-01T00:00:00Z
date_updated: 2026-07-16T08:43:40Z
day: '01'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1017/prm.2024.7
ec_funded: 1
external_id:
arxiv:
- '2203.02015'
isi:
- '001174907100001'
file:
- access_level: open_access
checksum: c5ec6e29aca2fb4533cb95fac409a0b2
content_type: application/pdf
creator: dernst
date_created: 2025-12-30T06:45:47Z
date_updated: 2025-12-30T06:45:47Z
file_id: '20878'
file_name: 2025_ProceedingsRoyalSocEdinburghA_Naskrecki.pdf
file_size: 477624
relation: main_file
success: 1
file_date_updated: 2025-12-30T06:45:47Z
fulldoi: https://doi.org/10.1017/prm.2024.7
has_accepted_license: '1'
intvolume: ' 155'
isi: 1
issue: '5'
keyword:
- Elliptic curves
- Néron models
- division polynomials
- height functions
- discrete valuation rings
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '10'
oa: 1
oa_version: Published Version
page: 1646-1660
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
call_identifier: H2020
grant_number: '101034413'
name: 'IST-BRIDGE: International postdoctoral program'
publication: 'Proceedings of the Royal Society of Edinburgh Section A: Mathematics'
publication_identifier:
eissn:
- 1473-7124
issn:
- 0308-2105
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Common valuations of division polynomials
tmp:
image: /images/cc_by.png
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)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 155
year: '2025'
...
---
OA_place: repository
OA_type: green
_id: '22196'
abstract:
- lang: eng
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."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Vivian Zieve
full_name: Kuperberg, Vivian Zieve
id: c3bac823-112d-11f0-a3f5-c264f852e697
last_name: Kuperberg
citation:
ama: Kuperberg VZ. On pseudo-polynomials divisible only by a sparse set of primes
and α-primary pseudo-polynomials. Journal of Number Theory. 2022;241:531-541.
doi:10.1016/j.jnt.2022.04.006
apa: Kuperberg, V. Z. (2022). On pseudo-polynomials divisible only by a sparse set
of primes and α-primary pseudo-polynomials. Journal of Number Theory. Elsevier.
https://doi.org/10.1016/j.jnt.2022.04.006
chicago: Kuperberg, Vivian Zieve. “On Pseudo-Polynomials Divisible Only by a Sparse
Set of Primes and α-Primary Pseudo-Polynomials.” Journal of Number Theory.
Elsevier, 2022. https://doi.org/10.1016/j.jnt.2022.04.006.
ieee: V. Z. Kuperberg, “On pseudo-polynomials divisible only by a sparse set of
primes and α-primary pseudo-polynomials,” Journal of Number Theory, vol.
241. Elsevier, pp. 531–541, 2022.
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.
mla: Kuperberg, Vivian Zieve. “On Pseudo-Polynomials Divisible Only by a Sparse
Set of Primes and α-Primary Pseudo-Polynomials.” Journal of Number Theory,
vol. 241, Elsevier, 2022, pp. 531–41, doi:10.1016/j.jnt.2022.04.006.
short: V.Z. Kuperberg, Journal of Number Theory 241 (2022) 531–541.
date_created: 2026-06-29T12:58:07Z
date_published: 2022-05-18T00:00:00Z
date_updated: 2026-07-14T11:08:14Z
day: '18'
doi: 10.1016/j.jnt.2022.04.006
extern: '1'
external_id:
arxiv:
- '2006.02527'
fulldoi: https://doi.org/10.1016/j.jnt.2022.04.006
intvolume: ' 241'
keyword:
- Pseudo-polynomials
- Chinese remainder theorem
- Ruzsa’s conjecture
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2006.02527
month: '05'
oa: 1
oa_version: Preprint
page: 531-541
publication: Journal of Number Theory
publication_identifier:
issn:
- 0022-314X
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: On pseudo-polynomials divisible only by a sparse set of primes and α-primary
pseudo-polynomials
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 241
year: '2022'
...