{
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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $p$ be a polynomial of degree $n$ with complex coefficients whose [[roots|019MA1VTR2]] all lie in the closed unit disc $\\set{z \\in \\C : \\abs{z} \\le 1}$.\nThen for each root $a$ of $p$ there is a root $b$ of the derivative $p'$ with $\\abs{a - b} \\le 1$.\n\n## Remarks\n\nThe roots of $p'$ are the [[critical points|0YTCMFB4HC]] of $p$.\nThe Gauss–Lucas theorem says that the critical points lie in the [[convex hull|0FQSDC2EQ2]] of the roots; this only gives the distance bound $2$.\nThe bound $1$ would be sharp: for $p(z) = z^n - 1$ and the root $a = 1$, the only critical point is $0$, at distance exactly $1$.\nThe conjecture has been proved for polynomials of degree at most $8$, by Brown and Xiang.\nTerence Tao proved it for all polynomials of sufficiently large degree $n$.\nThe general case is open.\n\n## Explanations\n\n### L1 human\n\nPlace the roots of a polynomial inside the unit disc.\nThe derivative of the polynomial has its own roots, which tend to sit among the roots of the polynomial.\nThe conjecture says that every root of the polynomial has a root of the derivative within distance $1$.\nIt is simple to state, yet it has resisted proof for decades.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
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    "history": {
      "first_stated": "1959",
      "posed_by": "Blagovest Sendov"
    },
    "kind": "conjecture",
    "msc": [
      "30C15",
      "30C10"
    ],
    "name": "Sendov's conjecture",
    "same_as": {
      "wikidata": "Q17037028",
      "wikipedia": "Sendov's conjecture"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Sendov%27s_conjecture"
      }
    ],
    "tag": "101H2VSX2R",
    "uses": [
      "019MA1VTR2",
      "0YTCMFB4HC",
      "0FQSDC2EQ2"
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  },
  "kind": "conjecture",
  "link": "https://math2.tamnd.com/node/101H2VSX2R/",
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      "target": "019MA1VTR2"
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      "target": "0YTCMFB4HC"
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      "target": "0FQSDC2EQ2"
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  "markdown": "https://math2.tamnd.com/node/101H2VSX2R.md",
  "msc": [
    "30C15",
    "30C10"
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  "name": "Sendov's conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/101/101H2VSX2R.md",
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  "tag": "101H2VSX2R",
  "uri": "m2://101H2VSX2R",
  "url": "https://math2.tamnd.com/wiki/sendovs-conjecture/",
  "writing": {
    "tag": "101H2VSX2R",
    "kind": "conjecture",
    "level": "W0",
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        "want": 1
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