{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "8EV392NDQD"
    },
    {
      "field": "uses",
      "target": "GA8BDBHP4M"
    }
  ],
  "body": "\n## Summary\n\nA central polynomial for the matrix ring $M_n(k)$ is a noncommutative polynomial whose values on $n \\times n$ matrices are always scalar matrices, but not always zero.\nTheir existence for every $n$ was proved independently by Formanek and by Razmyslov.\nCentral polynomials produce many central elements in [[polynomial identity rings|C0EDT81EGH]], and they are a main tool in the structure theory of these rings.\n\n## Explanations\n\n### L1 human\n\nThe values of most expressions in matrices, such as $xy - yx$, are ordinary matrices that are not multiples of the identity.\nA central polynomial is a special expression whose value is always a multiple of the identity matrix, whatever matrices one substitutes, and is sometimes nonzero.\nFor $2 \\times 2$ matrices, $(xy - yx)^2$ is such an expression.\nFor example, with $x = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ and $y = \\begin{pmatrix} 0 \u0026 0 \\\\ 1 \u0026 0 \\end{pmatrix}$ one gets $xy - yx = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 -1 \\end{pmatrix}$, whose square is the identity matrix.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "16R10",
      "16R20"
    ],
    "name": "Central polynomial",
    "same_as": {
      "wikidata": "Q18345072",
      "wikipedia": "Central polynomial"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Central_polynomial"
      }
    ],
    "tag": "667YZTHE96"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/667YZTHE96/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/667YZTHE96.md",
  "msc": [
    "16R10",
    "16R20"
  ],
  "name": "Central polynomial",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/667/667YZTHE96.md",
  "status": null,
  "tag": "667YZTHE96",
  "uri": "m2://667YZTHE96",
  "url": "https://tamnd.github.io/math2/wiki/central-polynomial/",
  "writing": {
    "tag": "667YZTHE96",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}