{
  "id": "2205.08825",
  "version": "v1",
  "published": "2022-05-18T09:42:57.000Z",
  "updated": "2022-05-18T09:42:57.000Z",
  "title": "Certain invariant algebraic sets in $S^p \\times S^q$",
  "authors": [
    "Joji Benny",
    "Soumen Sarkar"
  ],
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "Let $S_{p,q}$ be the hypersurface in $\\mathbb{R}^{p+q+1}$ defined by the following: $$ S_{p,q} := \\left\\lbrace (x_1,\\ldots,x_{p+1},y_1,\\ldots,y_q) \\in \\mathbb{R}^{p+q+1} \\big| \\left( \\sum_{i=1}^{p+1} x_i^2 - a^2 \\right)^2 + \\sum_{j=1}^q y_j^2 = 1 \\right\\rbrace,$$ where $a > 1$. We show that $S_{p,q}$ is homeomorphic to the product $S^p \\times S^q$. We consider the polynomial vector field $\\mathcal{X} = (P_1,...,P_{p+1},Q_1,...,Q_q)$ in $\\mathbb{R}^{p+q+1}$ which keeps $S_{p,q}$ invariant. Then we study the number of certain invariant algebraic subvarieties of $S_{p,q}$ for the vector field $\\mathcal{X}$ if either $p>1$ or $q>1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-18T09:42:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34A34",
      "34C14",
      "34C40",
      "34C45",
      "58J90"
    ],
    "keywords": [
      "invariant algebraic sets",
      "invariant algebraic subvarieties",
      "polynomial vector field",
      "homeomorphic",
      "hypersurface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}