{
  "abstract": "For an infinite field $K$ and an ideal $I\\subseteq K[X_1,\\dots,X_n]$, let $J(Z_K(I))$ be the ideal of polynomials vanishing on the $K$-rational zero locus of $I$. A Nullstellensatz over $K$ is an algebraic description of this closure operator on ideals. Hilbert's classical theorem provides such a description via the radical of $I$ over any algebraically closed field. In the present article, we introduce a unifying framework for Nullstellensätze over arbitrary rings using the notion of an $A$-radical, for $A$ is a subring of $K(X)$ containing $K[X]$. We show that a single canonical holomorphy ring gives a Nullstellensatz over every infinite field, recovering Hilbert's theorem when $K$ is algebraically closed. In addition, we give a new characterization of large fields (in the sense of Pop) via a particular Nullstellensatz: a perfect field is large exactly when the \\emph{rational} holomorphy ring $A_n(K)$--the intersection of all $K$-rational places of residue field exactly $K$--satisfies the corresponding Nullstellensatz. Finally we study the order structure of the class of rings $A$ for which the $A$-radical gives a Nullstellensatz. We show that over non-algebraically closed fields this class has no minimal element, showing there is no optimal Nullstellensatz over such fields.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21039",
  "author_total": 2,
  "authors": [
    {
      "name": "Pablo Cubides Kovacsics"
    },
    {
      "name": "Sebastián Uribe Rincón"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dhn0c2b11xaf6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21039",
  "fulltext_url": null,
  "id": "dhn0c2b11xaf6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21039",
  "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": "W7222556116",
  "pdf_url": "https://arxiv.org/pdf/2610.21039",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dhn0c2b11xaf6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21039",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21039"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Holomorphy rings and the Nullstellensatz over large fields",
  "title_text": "Holomorphy rings and the Nullstellensatz over large fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/dhn0c2b11xaf6",
  "venue": "arXiv",
  "versions": []
}