{
  "id": "2006.11260",
  "version": "v1",
  "published": "2020-06-19T17:48:45.000Z",
  "updated": "2020-06-19T17:48:45.000Z",
  "title": "Vectors of type II Hermite-Padé approximations and a new linear independence criterion",
  "authors": [
    "Raffaele Marcovecchio"
  ],
  "comment": "25 pages, 62 references",
  "categories": [
    "math.NT"
  ],
  "abstract": "We propose a linear independence criterion, and outline an application of it. Down to its simplest case, it aims at solving this problem: given three real numbers, typically as special values of analytic functions, how to prove that the $\\Q$-vector space spanned by $1$ and those three numbers has dimension at least 3, whenever we are unable to achieve full linear independence, by using simultaneous approximations, i.e. those usually arising from Hermite-Pad\\'e approximations of type II and their suitable generalizations. It should be recalled that approximations of type I and II are related, at least in principle: when the numerical application consists in specializing actual functional constructions of the two types, they can be obtained, one from the other, as explained in a well-known paper by K.Mahler \\cite{Mahler}. That relation is reflected in a relation between the asymptotic behavior of the approximations at the infinite place of $\\Q$. Rather interestingly, the two view-points split away regarding the asymptotic behaviors at finite places (i.e. primes) of $\\Q$, and this makes the use of type II more convenient for particular purposes. In addition, sometimes we know type II approximations to a given set of functions, for which type I approximations are not known explicitly. Our approach can be regarded as a dual version of the standard linear independence criterion, which goes back to Siegel.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-19T17:48:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J72"
    ],
    "keywords": [
      "asymptotic behavior",
      "achieve full linear independence",
      "standard linear independence criterion",
      "specializing actual functional constructions",
      "view-points split away"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}