{
  "abstract": "We develop a unified defect theory for embedded algebraic vector bundles which places several classical constructions within a common framework. The theory recovers tangential, dual, join, and secant defects. We first prove an effective numerical criterion characterizing defectivity by the vanishing of a bidegree. This also allows us to recover the exact defect from the vanishing pattern of the bidegrees, extending a theorem of Holme. Using van der Waerden's theorem on bidegrees, we obtain a formula that decomposes the geometric degree of an embedded vector bundle into contributions from the direction varieties of its general linear restrictions. This leads to a characterization of embedded algebraic vector bundles of minimal degree. As a main application, we establish the sharp universal bound $\\operatorname{deg}(TV) \\leq \\operatorname{deg}(V)^2$ for every smooth irreducible affine variety $V \\subseteq \\mathbb{A}^n$ and $TV\\subseteq \\mathbb{A}^{2n}$ its tangent bundle. Finally, under suitable regularity assumptions, we apply the quadratic bound to prolongation varieties of differential algebraic systems, obtaining uniform degree estimates and thereby providing a partial answer to an open problem in differential algebra posed by Pogudin.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27047",
  "author_total": 1,
  "authors": [
    {
      "name": "Leonardo Lanciano"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7893zg9w80gtw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27047",
  "fulltext_url": null,
  "id": "7893zg9w80gtw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27047",
  "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.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214206334",
  "pdf_url": "https://arxiv.org/pdf/2609.27047",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7893zg9w80gtw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27047",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27047"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Degrees and directional defects of embedded algebraic vector bundles",
  "title_text": "Degrees and directional defects of embedded algebraic vector bundles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/7893zg9w80gtw",
  "venue": "arXiv",
  "versions": []
}