{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "7AWZJNQGT2"
    },
    {
      "field": "uses",
      "target": "7AWZJNQGT2"
    },
    {
      "field": "defines",
      "target": "G3CKBSCM6A"
    }
  ],
  "body": "\n## Summary\n\nA compact set $K \\subseteq \\C^n$ is polynomially convex if every point outside $K$ can be separated from $K$ by a polynomial.\nThis means that some polynomial is larger in absolute value at the point than everywhere on $K$.\nThey are the natural sets for polynomial approximation.\nBy the [[Oka–Weil theorem|7AWZJNQGT2]], every function holomorphic near such a set is a uniform limit of polynomials on it.\nIn $\\C$ they are the compact sets with connected complement.\nFor $n \\ge 2$ the condition is much subtler, and it is not determined by the topology of $K$ alone.\n\n## Explanations\n\n### L1 human\n\nThe hull of a compact set $K$ collects the points where no polynomial beats its maximum over $K$.\nThe set is polynomially convex if the hull adds nothing.\nA closed disc in $\\C$ is polynomially convex, but a circle is not: by the maximum principle, its hull is the whole closed disc.\nEvery compact convex set in $\\C^n$ is polynomially convex, and so is every compact subset of $\\R^n \\subseteq \\C^n$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32E20"
    ],
    "name": "Polynomially convex set",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Function_of_several_complex_variables"
      }
    ],
    "tag": "AD3474PVHR"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/AD3474PVHR/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/AD3474PVHR.md",
  "msc": [
    "32E20"
  ],
  "name": "Polynomially convex set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AD3/AD3474PVHR.md",
  "status": null,
  "tag": "AD3474PVHR",
  "uri": "m2://AD3474PVHR",
  "url": "https://tamnd.github.io/math2/wiki/polynomially-convex-set/",
  "writing": {
    "tag": "AD3474PVHR",
    "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
      }
    ]
  }
}