@article{12311,
  abstract     = {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.},
  author       = {Naskręcki, Bartosz and Verzobio, Matteo},
  issn         = {1473-7124},
  journal      = {Proceedings of the Royal Society of Edinburgh Section A: Mathematics},
  keywords     = {Elliptic curves, Néron models, division polynomials, height functions, discrete valuation rings},
  number       = {5},
  pages        = {1646--1660},
  publisher    = {Cambridge University Press},
  title        = {{Common valuations of division polynomials}},
  doi          = {10.1017/prm.2024.7},
  volume       = {155},
  year         = {2025},
}

@article{22196,
  abstract     = {We explore two questions about pseudo-polynomials, which
are functions f : N → Z such that k divides f(n + k) −
f(n) for all n, k. First, for certain arbitrarily sparse sets R, we
construct pseudo-polynomials f with p|f(n) for some n only if
p ∈ R. This implies that not all pseudo-polynomials satisfy an
assumption of a recent paper of Kowalski and Soundararajan.
We also consider α-primary pseudo-polynomials, where the
pseudo-polynomial condition is only required for k lying in
a set of primes of density α. We show that if an α-primary
pseudo-polynomial is O(e(β−)n), where β = √7
3 − 1
6 ≈ 0.715,
then it is a polynomial.},
  author       = {Kuperberg, Vivian Zieve},
  issn         = {0022-314X},
  journal      = {Journal of Number Theory},
  keywords     = {Pseudo-polynomials, Chinese remainder theorem, Ruzsa’s conjecture},
  pages        = {531--541},
  publisher    = {Elsevier},
  title        = {{On pseudo-polynomials divisible only by a sparse set of primes and α-primary pseudo-polynomials}},
  doi          = {10.1016/j.jnt.2022.04.006},
  volume       = {241},
  year         = {2022},
}