{
  "abstract": "If $f$ is a tuple of functions satisfying an algebraic ODE and $P\\in{\\mathbb C}(x)[f]$, 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 $\\operatorname{deg}_x P,\\operatorname{deg}_f P$. 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 $\\mathbb C(x)$ is replaced by a number field $K$ and $\\operatorname{deg}_x P$ is replaced by the logarithmic height $\\operatorname{h}(P)$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24148",
  "author_total": 2,
  "authors": [
    {
      "name": "Gal Binyamini"
    },
    {
      "name": "Yuval Salant"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3ywhe5sr8qt8v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24148",
  "fulltext_url": null,
  "id": "3ywhe5sr8qt8v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24148",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213995518",
  "pdf_url": "https://arxiv.org/pdf/2609.24148",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3ywhe5sr8qt8v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24148",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24148"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp zero estimates for trajectories of polynomial vector fields",
  "title_text": "Sharp zero estimates for trajectories of polynomial vector fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/3ywhe5sr8qt8v",
  "venue": "arXiv",
  "versions": []
}