{
  "id": "1310.8563",
  "version": "v4",
  "published": "2013-10-31T15:55:45.000Z",
  "updated": "2013-12-15T14:36:04.000Z",
  "title": "The images of non-commutative polynomials evaluated on $2\\times 2$ matrices over an arbitrary field",
  "authors": [
    "Sergey Malev"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set $sl_n(K)$ of matrices of trace $0$, or all of $M_n(K)$. This conjecture was proved for $n=2$ when $K$ is closed under quadratic extensions. In this paper the conjecture is verified for $K=\\mathbb{R}$ and $n=2$, also for semi-homogeneous polynomials $p$, with a partial solution for an arbitrary field $K$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-12-15T14:36:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arbitrary field",
      "non-commutative polynomials",
      "conjecture",
      "multilinear polynomial",
      "scalar matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.8563M"
    }
  }
}