{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $f, g \\in \\C\\{z_0, \\dots, z_n\\}$ be [[convergent power series|EVPZ7ZG1HB]] without repeated factors, with $f(0) = g(0) = 0$.\nThe *multiplicity* $\\operatorname{mult}_0 f$ is the least degree of a nonzero homogeneous term in the power series expansion of $f$.\nSuppose that there is a [[homeomorphism|0QT00PB5G2]] $\\phi$ between neighbourhoods of $0$ in $\\C^{n+1}$ with $\\phi(0) = 0$.\nSuppose that $\\phi$ maps the [[germ|VZXF7RNFKS]] of $f^{-1}(0)$ at $0$ onto the germ of $g^{-1}(0)$ at $0$.\nThen $\\operatorname{mult}_0 f = \\operatorname{mult}_0 g$.\n\n## Remarks\n\nZariski asked this question in his 1971 paper \"Some open questions in the theory of singularities\".\nIt is open in general.\nIt holds for plane curves, $n = 1$, where the embedded topological type determines the Puiseux characteristics of the branches and hence the multiplicity.\nThe multiplicity of $f$ equals the [[multiplicity of the local ring|JX20MNDDXT]] $\\C\\{z_0, \\dots, z_n\\} / (f)$, so the conjecture asks whether this algebraic invariant is determined by the embedded topology.\nThe Milnor number, by contrast, is known to be a topological invariant in this sense.\n\n## Explanations\n\n### L1 human\n\nThe multiplicity of a hypersurface at a point counts how many sheets meet there: a smooth point has multiplicity $1$, two crossing lines have multiplicity $2$.\nZariski asked whether this number can be read off from the shape of the hypersurface near the point, as seen by topology, without using its algebra.\nFor curves in the plane the answer is yes, but in higher dimensions the question is still unanswered.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "42V5XGZS4F",
      "JX20MNDDXT"
    ],
    "history": {
      "first_stated": "1971",
      "posed_by": "Oscar Zariski"
    },
    "kind": "conjecture",
    "msc": [
      "32S05",
      "32S15",
      "14B05"
    ],
    "name": "Zariski's multiplicity conjecture",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Singularity_theory"
      }
    ],
    "tag": "ZTTZVZWMK9",
    "uses": [
      "EVPZ7ZG1HB",
      "VZXF7RNFKS",
      "0QT00PB5G2"
    ]
  },
  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/ZTTZVZWMK9/",
  "links": [
    {
      "field": "about",
      "target": "42V5XGZS4F"
    },
    {
      "field": "about",
      "target": "JX20MNDDXT"
    },
    {
      "field": "uses",
      "target": "EVPZ7ZG1HB"
    },
    {
      "field": "uses",
      "target": "VZXF7RNFKS"
    },
    {
      "field": "uses",
      "target": "0QT00PB5G2"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/ZTTZVZWMK9.md",
  "msc": [
    "32S05",
    "32S15",
    "14B05"
  ],
  "name": "Zariski's multiplicity conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZTT/ZTTZVZWMK9.md",
  "status": null,
  "tag": "ZTTZVZWMK9",
  "uri": "m2://ZTTZVZWMK9",
  "url": "https://tamnd.github.io/math2/wiki/zariskis-multiplicity-conjecture/",
  "writing": {
    "tag": "ZTTZVZWMK9",
    "kind": "conjecture",
    "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
      }
    ]
  }
}