{
  "id": "1003.1746",
  "version": "v2",
  "published": "2010-03-08T22:36:30.000Z",
  "updated": "2019-04-06T12:07:54.000Z",
  "title": "Invariants of Relative Right and Contact Equivalences",
  "authors": [
    "Imran Ahmed",
    "Maria Aparecida Soares Ruas"
  ],
  "comment": "11 pages. arXiv admin note: substantial text overlap with arXiv:0909.5429",
  "journal": "Houston Journal of Mathematics, 37 (2011), no. 3, 773-786",
  "categories": [
    "math.AG"
  ],
  "abstract": "We study holomorphic function germs under equivalence relations that preserve an analytic variety. We show that two quasihomogeneous polynomials, not necessarily with isolated singularities, having isomorphic relative Milnor algebras are relative right equivalent. Under the condition that the module of vector fields tangent to the variety is finitely generated, we also show that the relative Tjurina algebra is a complete invariant for the classification of arbitrary function germs with respect to the relative contact equivalence. This is the relative version of a well known result by Mather and Yau.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-03-08T22:36:30.000Z",
      "title": "Invariants of $R_v$-Equivalence",
      "abstract": "We recall Mather's Lemma providing effective necessary and sufficient conditions for a connected submanifold to be contained in an orbit. In Theorem 5.2 we show that two quasihomogeneous polynomials $f$ and $g$ having isomorphic relative Milnor algebras $M_v(f)$ and $M_v(g)$ are $R_v$-equivalent. In Theorem 5.4 we prove that two complex-analytic hypersurfaces, one is quasihomogeneous and other is arbitrary, are determined by isomorphism of Jacobean ideals. The Example of Gaffney and Hauser, in 1985, suggests us that we can not extend our results for arbitrary analytic germs.",
      "comment": "11 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2019-04-06T12:07:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E05",
      "32S30",
      "14L30",
      "14J17",
      "16W22"
    ],
    "keywords": [
      "invariants",
      "equivalence",
      "recall mathers lemma",
      "arbitrary analytic germs",
      "isomorphic relative milnor algebras"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.1746A"
    }
  }
}