{
  "abstract": "We analyze the behavior of systems of algebraic differential equations when considered as systems of difference-differential equations, with special emphasis on systems which define minimal sets relative to the theory $\\operatorname{DCF}$ of differentially closed fields of characteristic zero with $m$ distinguished commuting derivations. We show that if $p$ is a minimal type relative to $\\operatorname{DCF}$ defined by a system of algebraic partial differential equations and inequations and the forking geometry on $p$ is trivial (also called ``disintegrated''), then $p$ remains minimal when regarded as a partial type relative to the theory $\\operatorname{DCFA}$ of difference-differentially closed fields of characteristic zero with $m$ commuting derivations in the sense that every completion has $\\operatorname{SU}$-rank one. We illustrate this theorem by describing in detail the possible difference-differential equations consistent with differential equations of the form $y' = y^3 + a_2 y^2 + a_1 y + a_0$ where $a_0$, $a_1$, and $a_2$ are differential constants.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22317",
  "author_total": 2,
  "authors": [
    {
      "name": "Wei Li"
    },
    {
      "name": "Thomas Scanlon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eyr8jzc904m7n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22317",
  "fulltext_url": null,
  "id": "eyr8jzc904m7n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22317",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7222586854",
  "pdf_url": "https://arxiv.org/pdf/2610.22317",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eyr8jzc904m7n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.22317",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22317"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Differential equations as difference-differential equations",
  "title_text": "Differential equations as difference-differential equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/eyr8jzc904m7n",
  "venue": "arXiv",
  "versions": []
}