{
  "abstract": "We establish a finite-basis theorem for radical well-mixed difference ideals in a difference polynomial ring with one endomorphism over an arbitrary difference field. The result extends to every finitely difference-generated difference algebra over such a field and is equivalent to the ascending chain condition for radical well-mixed difference ideals, hence resolving a long-standing open problem in difference algebra. The proof proceeds in two stages. Over an inversive difference coefficient field, we establish a uniform inverse-transform bound relating the perfect closure of a finite system to its radical well-mixed closure. This uses Noetherian induction in a fixed truncated polynomial ring, together with generic freeness and going-down for flat ring homomorphisms. We then prove that the radical well-mixed closure of a system over a difference field $K$ is the contraction of its radical well-mixed closure over the inversive closure of $K$. This contraction result follows from successive prime lifting along flat extensions and removes the inversiveness assumption without requiring the coefficient-field extension to be finitely generated.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34389",
  "author_total": 2,
  "authors": [
    {
      "name": "Wei Li"
    },
    {
      "name": "Jie Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3vv5byeqvdmf1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34389",
  "fulltext_url": null,
  "id": "3vv5byeqvdmf1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34389",
  "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.AC"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "1703|Polynomial and algebraic computation"
    ]
  ],
  "openalex": "W7223959167",
  "pdf_url": "https://arxiv.org/pdf/2610.34389",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3vv5byeqvdmf1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34389",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34389"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The ascending chain condition for radical well-mixed difference ideals",
  "title_text": "The ascending chain condition for radical well-mixed difference ideals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/3vv5byeqvdmf1",
  "venue": "arXiv",
  "versions": []
}