{
  "id": "1306.3314",
  "version": "v2",
  "published": "2013-06-14T07:13:03.000Z",
  "updated": "2013-09-13T04:39:18.000Z",
  "title": "Jacobian Conjecture in two dimension",
  "authors": [
    "Dosang Joe"
  ],
  "comment": "This paper has been withdrawn by the author due to a crucial error in the proposition 1",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "abstract": "Let $(P, Q)$ be a pair of Jacobian polynomials. We can show that $ <P, Q>+l+2g(P)-2= 0= <P, [P,Q]>$, where $<f, g>$ is the intersection number of $f, g\\in \\CC[x, y]$ in the affine plane, $l$ is the number of branch at point at infinity and $g(P)$ is the geometric genus of affine curve defined by $P$. Hence we can show that every Jacobian polynomial defines a smooth rational curve with one point at infinity. It is sufficient to fix the Jacobian conjecture in two dimension by the Abhyankar theorem or the Abhyankar-Moh-Suzuki theorem.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-09-13T04:39:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14R15",
      "37D15"
    ],
    "keywords": [
      "jacobian conjecture",
      "smooth rational curve",
      "jacobian polynomial defines",
      "affine plane",
      "abhyankar-moh-suzuki theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.3314J"
    }
  }
}