{
  "id": "1601.01361",
  "version": "v1",
  "published": "2016-01-07T00:42:04.000Z",
  "updated": "2016-01-07T00:42:04.000Z",
  "title": "Lattices over Polynomial Rings and Applications to Function Fields",
  "authors": [
    "Jens-Dietrich Bauch"
  ],
  "comment": "32 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "This paper deals with lattices $(L,\\Vert~\\Vert)$ over polynomial rings, where $L$ is a finitely generated module over $k[t]$, the polynomial ring over the field $k$ in the indeterminate $t$, and $\\Vert~\\Vert$ is a discrete real-valued length function on $L\\otimes_{k[t]}k(t)$. A reduced basis of $(L,\\Vert~\\Vert)$ is a basis of $L$ whose vectors attain the successive minima of $(L,\\Vert~\\Vert)$. We develop an algorithm which transforms any basis of $L$ into a reduced basis of $(L,\\Vert~\\Vert)$. By identifying a divisor $D$ of an algebraic function field with a lattice $(L,\\Vert~\\Vert)$ over a polynomial ring, this reduction algorithm can be addressed to the computation of the Riemann-Roch space of $D$ and the successive minima of $(L,\\Vert~\\Vert)$, without the use of any series expansion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-07T00:42:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial ring",
      "applications",
      "discrete real-valued length function",
      "successive minima",
      "reduced basis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160101361B"
    }
  }
}