{
  "abstract": "Let $R$ be a real closed field and let $X_Σ$ be a smooth projective split toric variety. For an explicitly chosen basis rigidification $\\mathfrak r$ of its Picard grading, we prove that the pair $(Σ,\\mathfrak r)$ determines the resulting total Chow--Witt ring and give a terminating finite algorithm for all groups, products, twists, and forgetful maps. The ring is identified with an explicit fibre product of the Picard-graded diagonal $\\mathbf{I}$-cohomology ring and the integral inverse images of twisted Bockstein kernels over the mod-$2$ Chow ring. Using real cycle classes, we identify the first factor with the cohomology of the real toric variety with all sign local systems. We construct the mod-$2$ cycle map on invariant divisors as explicit deck-transition cocycles and obtain finite signed boundary and product matrices directly from the fan.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26082",
  "author_total": 3,
  "authors": [
    {
      "name": "Haoyang Liu"
    },
    {
      "name": "Tianle Liu"
    },
    {
      "name": "Guorui Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dessgm3gm0dpp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26082",
  "fulltext_url": null,
  "id": "dessgm3gm0dpp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26082",
  "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.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214105627",
  "pdf_url": "https://arxiv.org/pdf/2609.26082",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dessgm3gm0dpp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26082",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26082"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Fan Algorithm for Chow-Witt Rings of Smooth Projective Toric Varieties",
  "title_text": "A Fan Algorithm for Chow-Witt Rings of Smooth Projective Toric Varieties",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/dessgm3gm0dpp",
  "venue": "arXiv",
  "versions": []
}