{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$, write $z' = (z_1, \\dots, z_{n-1})$, and let $\\mathcal{O}_{n-1} = \\C\\{z'\\}$ be the [[ring of convergent power series|EVPZ7ZG1HB]] in $n - 1$ variables.\nLet $d \\ge 0$ be an integer.\nA *Weierstrass polynomial* of degree $d$ in $z_n$ is a [[monic polynomial|0C3QBCR46M]]\n$$P(z', z_n) = z_n^d + a_{d-1}(z')\\, z_n^{d-1} + \\cdots + a_1(z')\\, z_n + a_0(z')$$\nwith coefficients $a_0, \\dots, a_{d-1} \\in \\mathcal{O}_{n-1}$ such that $a_j(0) = 0$ for every $j$.\n\n## Remarks\n\nEquivalently, $P \\in \\mathcal{O}_{n-1}[z_n]$ is monic of degree $d$ and $P(0, z_n) = z_n^d$.\nThe only Weierstrass polynomial of degree $0$ is $P = 1$.\nExample: $P = z_2^3 - z_1 z_2 + z_1^2$ is a Weierstrass polynomial of degree $3$ in $z_2$, since $a_2 = 0$, $a_1 = -z_1$ and $a_0 = z_1^2$ all vanish at $0$.\nNon-example: $Q = z_2^2 + z_2 + z_1$ is monic in $z_2$, but its coefficient $a_1 = 1$ does not vanish at $0$; indeed $Q(0, z_2) = z_2^2 + z_2 \\ne z_2^2$.\nThe same definition is used with $\\mathcal{O}_{n-1}$ replaced by the ring of holomorphic functions on a neighbourhood of $0 \\in \\C^{n-1}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "SDCKNWSWNM",
    "kind": "definition",
    "msc": [
      "32B05",
      "13J07"
    ],
    "name": "Weierstrass polynomial",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Weierstrass_preparation_theorem"
      }
    ],
    "tag": "RD7DVC2NJ6",
    "uses": [
      "EVPZ7ZG1HB",
      "0C3QBCR46M"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/RD7DVC2NJ6/",
  "links": [
    {
      "field": "defines",
      "target": "SDCKNWSWNM"
    },
    {
      "field": "uses",
      "target": "EVPZ7ZG1HB"
    },
    {
      "field": "uses",
      "target": "0C3QBCR46M"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/RD7DVC2NJ6.md",
  "msc": [
    "32B05",
    "13J07"
  ],
  "name": "Weierstrass polynomial",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RD7/RD7DVC2NJ6.md",
  "status": null,
  "tag": "RD7DVC2NJ6",
  "uri": "m2://RD7DVC2NJ6",
  "url": "https://tamnd.github.io/math2/wiki/weierstrass-polynomial-definition/",
  "writing": {
    "tag": "RD7DVC2NJ6",
    "kind": "definition",
    "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
      }
    ]
  }
}