{
  "id": "1812.10151",
  "version": "v1",
  "published": "2018-12-25T18:51:28.000Z",
  "updated": "2018-12-25T18:51:28.000Z",
  "title": "Expansions of real closed fields which introduce no new smooth functions",
  "authors": [
    "Pantelis E. Eleftheriou",
    "Alex Savatovsky"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We prove the following theorem: let $\\widetilde{\\mathcal R}$ be an expansion of the real field $\\overline{\\mathbb R}$, such that every definable set (I) is a uniform countable union of semialgebraic sets, and (II) contains a \"semialgebraic chunk\". Then every definable smooth function $f:X\\subseteq \\mathbb R^n\\to \\mathbb R$ with open semialgebraic domain is semialgebraic. Conditions (I) and (II) hold for various d-minimal expansions $\\widetilde{\\mathcal R} = \\langle \\overline{\\mathbb R}, P\\rangle$ of the real field, such as when $P=2^\\mathbb Z$, or $P\\subseteq \\mathbb R$ is an iteration sequence. A generalization of the theorem to d-minimal expansions $\\widetilde{\\mathcal R}$ of $\\mathbb R_{an}$ fails. On the other hand, we prove our theorem for expansions$\\widetilde{\\mathcal R}$ of arbitrary real closed fields. Moreover, its conclusion holds for certain structures with d-minimal open core, such as $\\langle \\overline{\\mathbb R}, \\mathbb R_{alg}, 2^\\mathbb Z\\rangle$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-25T18:51:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C64"
    ],
    "keywords": [
      "real field",
      "d-minimal expansions",
      "arbitrary real closed fields",
      "d-minimal open core",
      "open semialgebraic domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}