{
  "id": "2403.17478",
  "version": "v1",
  "published": "2024-03-26T08:15:45.000Z",
  "updated": "2024-03-26T08:15:45.000Z",
  "title": "C-minimal fields have the exchange property",
  "authors": [
    "Will Johnson"
  ],
  "comment": "41 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We show that C-minimal fields (i.e., C-minimal expansions of ACVF) have the exchange property, answering a question of Haskell and Macpherson. Additionally, we strengthen some theorems of Cubides Kovacsics and Delon on C-minimal fields. First, we show that definably complete C-minimal fields of characteristic 0 have generic differentiability. Second, we show that if the induced structure on the residue field is a pure ACF, then polynomial boundedness holds. In fact, polynomial boundedness can only fail if there are unexpected definable automorphisms of the multiplicative group of the residue field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-26T08:15:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C60"
    ],
    "keywords": [
      "exchange property",
      "residue field",
      "definably complete c-minimal fields",
      "polynomial boundedness holds",
      "generic differentiability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}