{
  "abstract": "We develop a complex-analytic approach to the model theory of the (real) unrestricted exponential. As a consequence we derive sharp forms of many of the foundational results for the structure ${\\mathbb R}^\\text{RE}_{\\exp}$ (and more general structures). In particular we establish sharp o-minimality, a sharp form of Wilkie's theorem of the complement, a sharp form of Wilkie's conjecture and a sharp form of piecewise definability by terms. Our approach is based on a complexification of the LE-preparation theorem of Lion--Rolin. We also develop a parallel complex theory for ${\\mathbb R}_\\text{an,exp}$, proving for example that the rational points of height $H$ on a nowhere-dense definable set can be interpolated by an algebraic hypersurface of degree $\\text{poly}(\\log H)$. This generalizes a theorem of Cluckers--Pila--Wilkie who proved the same statement for ${\\mathbb R}_\\text{an}^\\text{pow}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20668",
  "author_total": 3,
  "authors": [
    {
      "name": "Gal Binyamini"
    },
    {
      "name": "Oded Carmon"
    },
    {
      "name": "Dmitry Novikov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3k918ec64wnhx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20668",
  "fulltext_url": null,
  "id": "3k918ec64wnhx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20668",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7213598826",
  "pdf_url": "https://arxiv.org/pdf/2609.20668",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3k918ec64wnhx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20668",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20668"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Logarithmic--exponential preparation in sharply o-minimal structures",
  "title_text": "Logarithmic--exponential preparation in sharply o-minimal structures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/3k918ec64wnhx",
  "venue": "arXiv",
  "versions": []
}