{
  "id": "math/0306240",
  "version": "v1",
  "published": "2003-06-16T17:34:27.000Z",
  "updated": "2003-06-16T17:34:27.000Z",
  "title": "Bounds and definability in polynomial rings",
  "authors": [
    "Matthias Aschenbrenner"
  ],
  "comment": "36 pages",
  "categories": [
    "math.AC",
    "math.LO"
  ],
  "abstract": "We study questions around the existence of bounds and the dependence on parameters for linear-algebraic problems in polynomial rings over rings of an arithmetic flavor.In particular, we show that the module of syzygies of polynomials $f_1,...,f_n\\in R[X_1,...,X_N]$ with coefficients in a Pr\\\"ufer domain $R$ can be generated by elements whose degrees are bounded by a number only depending on $N$, $n$ and the degree of the $f_j$. This implies that if $R$ is a B\\'ezout domain, then the generators can be parametrized in terms of the coefficients of $f_1,...,f_n$ using the ring operations and a certain division function, uniformly in $R$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-06-16T17:34:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13D02",
      "13F05",
      "13L05"
    ],
    "keywords": [
      "polynomial rings",
      "definability",
      "arithmetic flavor",
      "linear-algebraic problems",
      "coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......6240A"
    }
  }
}