{
  "abstract": "We study the analytification and tropicalization of algebraic varieties over valued fields via Jun's spaces of hyperfield points. Our main new objects are a family of power-tropical hyperfields $H_n$ and their inverse limit $H$, recording at each level the class of a leading term modulo $n$-th powers of the residue field. They yield an enhanced tropicalization of the associated space $X(H)$, strictly finer than the Berkovich one exactly when the multiplicative group of the residue field is non-divisible: it specialises to the Berkovich analytification when the residue field is algebraically closed, to the real analytification of Jell-Scheiderer-Yu when it is real closed, and to a new $p$-adic analytification when it is $p$-adically closed. Over henselian valued fields of equicharacteristic $0$ we then construct a model-theoretic avatar of $X(H)$: the space $\\widehat{X}_T$ of definable types orthogonal to the value group, which sharpens the geometry in the spirit of Hrushovski and Loeser and recovers their space over algebraically closed valued fields. We show that $\\widehat{X}_T$ is strictly pro-definable, satisfies a model-theoretic Payne theorem, and definable compactness in such spaces satisfies a Heine-Borel type characterization. As an application, we analytify $p$-adic semi-algebraic sets and prove a strong density theorem for their tropicalizations, the exact $p$-adic counterpart of the real one of Jell-Scheiderer-Yu.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30762",
  "author_total": 3,
  "authors": [
    {
      "name": "Pablo Cubides Kovacsics"
    },
    {
      "name": "Mariana Vicaría"
    },
    {
      "name": "Jinhe Ye"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ajyrk7ccb1pj6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30762",
  "fulltext_url": null,
  "id": "ajyrk7ccb1pj6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30762",
  "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"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223259768",
  "pdf_url": "https://arxiv.org/pdf/2610.30762",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ajyrk7ccb1pj6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30762",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30762"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Analytification and tropicalization via hyperfields and definable types",
  "title_text": "Analytification and tropicalization via hyperfields and definable types",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/ajyrk7ccb1pj6",
  "venue": "arXiv",
  "versions": []
}