{
  "id": "2406.13729",
  "version": "v1",
  "published": "2024-06-19T17:48:50.000Z",
  "updated": "2024-06-19T17:48:50.000Z",
  "title": "On the Saito number of plane curves",
  "authors": [
    "Yohann Genzmer",
    "Marcelo E. Hernandes"
  ],
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "In this work we study the \\emph{Saito number} of a plane curve and we present a method to determine the minimal Saito number for plane curves in a given equisingularity class, that gives rise to an actual algorithm. In particular situations, we also provide various formulas for this number. In addition, if $\\nu_0$ and $\\nu_1$ are two coprime positive integers and $N>0$ then we show that for any $1\\leq k\\leq \\left [\\frac{N\\nu_0}{2}\\right ]$ there exits a plane curve equisingular to the curve $$y^{N\\nu_0}-x^{N\\nu_1}=0$$ such that its Saito number is precisely $k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-19T17:48:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H20",
      "32S65",
      "14B05"
    ],
    "keywords": [
      "minimal saito number",
      "plane curve equisingular",
      "equisingularity class",
      "actual algorithm",
      "coprime positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}