{
  "id": "1602.05732",
  "version": "v1",
  "published": "2016-02-18T09:47:18.000Z",
  "updated": "2016-02-18T09:47:18.000Z",
  "title": "Deformations of weighted homogeneous polynomials with line singularities and equimultiplicity",
  "authors": [
    "Christophe Eyral",
    "Maria Aparecida Soares Ruas"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Consider a family $\\{f_t\\}$ of complex polynomial functions with line singularities, and assume that $f_0$ is weighted homogeneous. We show that if the $1$-st L\\^e number and the $1$-st polar number of $f_t$ at $\\mathbf{0}$ are independent of $t$, then the family $\\{f_t\\}$ is equimultiple. In particular, $\\{f_t\\}$ is equimultiple if the $1$-st polar number of $f_t$ at $\\mathbf{0}$ and the local ambient topology of the hypersurface $f_t^{-1}(0)$ at $\\mathbf{0}$ are independent of $t$. This partially answers the famous Zariski multiplicity conjecture for this special class of non-isolated singularities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-18T09:47:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32S15",
      "32S25",
      "32S05"
    ],
    "keywords": [
      "line singularities",
      "weighted homogeneous polynomials",
      "st polar number",
      "deformations",
      "equimultiplicity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160205732E"
    }
  }
}