{
  "id": "math/0601336",
  "version": "v2",
  "published": "2006-01-13T15:18:15.000Z",
  "updated": "2006-09-08T20:03:37.000Z",
  "title": "Zeta Functions for Analytic Mappings, Log-principalization of Ideals, and Newton Polyhedra",
  "authors": [
    "Willem Veys",
    "W. A. Zuniga-Galindo"
  ],
  "comment": "Some typos were corrected. To appear in Trans. Amer. Math. Soc",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "In this paper we provide a geometric description of the possible poles of the Igusa local zeta function associated to an analytic mapping and a locally constant function, in terms of a log-principalizaton of an ideal naturally attached to the mapping. Typically our new method provides a much shorter list of possible poles compared with the previous methods. We determine the largest real part of the poles of the Igusa zeta function, and then as a corollary, we obtain an asymptotic estimation for the number of solutions of an arbitrary system of polynomial congruences in terms of the log-canonical threshold of the subscheme given by the ideal attached to the mapping. We associate to an analytic mapping a Newton polyhedron and a new notion of non-degeneracy with respect to it. By constructing a log-principalization, we give an explicit list for the possible poles of the Igusa zeta function associated to a non-degenerate mapping.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-09-08T20:03:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11S40",
      "11D79",
      "14M25",
      "32S45"
    ],
    "keywords": [
      "newton polyhedron",
      "analytic mapping",
      "igusa zeta function",
      "log-principalization",
      "igusa local zeta function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1336V"
    }
  }
}