Sharp zero estimates for trajectories of polynomial vector fields

@misc{binyamini2026sharp,
  title = {{Sharp zero estimates for trajectories of polynomial vector fields}},
  author = {Gal Binyamini and Yuval Salant},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24148},
  eprint = {2609.24148},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24148},
}

TY  - UNPB
TI  - Sharp zero estimates for trajectories of polynomial vector fields
AU  - Gal Binyamini
AU  - Yuval Salant
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24148
UR  - https://arxiv.org/abs/2609.24148
AB  - If f is a tuple of functions satisfying an algebraic ODE and P∈ℂ(x)[f], it is common in applications to transcendental number theory to consider upper bounds for the order of zero of P(x,f) at a given point in terms of deg_x P,deg_f P. Nesterenko introduced a condition known as the D-property, which holds in many applications, and proved essentially optimal bounds under this condition. We prove a global form of this result, where the field ℂ(x) is replaced by a number field K and deg_x P is replaced by the logarithmic height h(P). Under an analogous global D-property, we prove essentially optimal bounds for the number of zeroes, counted with multiplicities, in a fixed compact set. Applications of this result to point-counting theorems are developed in a separate joint paper with Hirata-Kohno and Kawashima.
ER  -