{
  "id": "2007.08280",
  "version": "v1",
  "published": "2020-07-16T12:14:37.000Z",
  "updated": "2020-07-16T12:14:37.000Z",
  "title": "Exponential periods and o-minimality I",
  "authors": [
    "Johan Commelin",
    "Philipp Habegger",
    "Annette Huber"
  ],
  "comment": "45 pages, comments welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\alpha \\in \\mathbb{C}$ be an exponential period. This is the first part of a pair of papers where we show that the real and imaginary part of $\\alpha$ are up to signs volumes of sets definable in the o-minimal structure generated by $\\mathbb{Q}$, the real exponential function and ${\\sin}|_{[0,1]}$. This is a weaker analogue of the precise characterisation of ordinary periods as numbers whose real and imaginary part are up to signs volumes of $\\mathbb{Q}$-semi-algebraic sets. Furthermore, we define a notion of naive exponential periods and compare it to the existing notions using cohomological methods. This points to a relation between the theory of periods and o-minimal structures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-16T12:14:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G35",
      "14F25",
      "14F40",
      "14P10",
      "03C64"
    ],
    "keywords": [
      "imaginary part",
      "signs volumes",
      "o-minimal structure",
      "o-minimality",
      "real exponential function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}