{
  "id": "2312.13865",
  "version": "v1",
  "published": "2023-12-21T14:02:08.000Z",
  "updated": "2023-12-21T14:02:08.000Z",
  "title": "Images of polynomial maps with constants",
  "authors": [
    "Saikat Panja",
    "Prachi Saini",
    "Anupam Singh"
  ],
  "comment": "Preliminary version; 34 pp;",
  "categories": [
    "math.GR",
    "math.RA"
  ],
  "abstract": "Let $K$ be an algebraically closed field and $\\mathrm{M}(2,K)$ be the $2\\times 2$ matrix algebra over $K$ and $\\mathrm{GL}(2,K)$ be the invertible elements in $\\mathrm{M}(2,K)$. We explore the image of polynomials with constants, namely from the free algebra $\\mathrm{M}(2,K)\\langle x, y\\rangle$. In this article, we compute the images of the polynomial maps given by (a) generalized sum of powers $Ax^{k_1} + By^{k_2}$ and (b) generalized commutator map $Axy -Byx$, where $A$, $B$ are non-zero elements of $\\mathrm{M}(2,K)$. We compute this in the first case by fixing a simultaneous conjugate pair for $A, B$ and it turns out that it is surjective in most of the cases. In the second case, we show that the image of the map is always a vector space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-21T14:02:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16S50",
      "11P05"
    ],
    "keywords": [
      "polynomial maps",
      "simultaneous conjugate pair",
      "matrix algebra",
      "generalized commutator map",
      "vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}