{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "06CTFY2DEZ"
    },
    {
      "field": "uses",
      "target": "0YKA9J3XEE"
    },
    {
      "field": "uses",
      "target": "0ZMY8441GW"
    }
  ],
  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $n \\ge 1$ be an integer.\nFor $\\alpha = (\\alpha_1, \\dots, \\alpha_n) \\in \\mathbb{N}^n$, the monomial $x^\\alpha$ is the formal product $x_1^{\\alpha_1} \\cdots x_n^{\\alpha_n}$.\nA polynomial in $x_1, \\dots, x_n$ over $R$ is a finite sum $\\sum_\\alpha c_\\alpha x^\\alpha$ with $c_\\alpha \\in R$.\nTwo polynomials are equal if and only if they have the same coefficient $c_\\alpha$ for each $\\alpha$.\nThe ring $R[x_1, \\dots, x_n]$ is the set of these polynomials, with coefficientwise addition and the product\n$$\\Big(\\sum_\\alpha c_\\alpha x^\\alpha\\Big) \\Big(\\sum_\\beta d_\\beta x^\\beta\\Big) = \\sum_\\gamma \\Big(\\sum_{\\alpha + \\beta = \\gamma} c_\\alpha d_\\beta\\Big) x^\\gamma.$$\n\n## Remarks\n\nThere is a natural isomorphism $R[x_1, \\dots, x_n] \\cong R[x_1, \\dots, x_{n-1}][x_n]$. So the ring can also be built by repeated use of the [[polynomial ring|0Q9EZS90GY]] in one variable.\nThe same construction works for any set of variables, with exponent maps that are zero at all but finitely many variables.\nThe polynomial $x_1 x_2 + x_3^2$ is an element of $\\mathbb{Z}[x_1, x_2, x_3]$.\n\n## Explanations\n\n### L1 human\n\nA polynomial in several variables is a finite sum of terms like $5 x^2 y z^3$.\nYou add and multiply such polynomials with the usual rules of algebra.\nEach term is a coefficient times a product of powers of the variables.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "07GNX8GCYP",
    "kind": "definition",
    "msc": [
      "13F20"
    ],
    "name": "Polynomials in several variables",
    "sources": [
      {
        "locator": "Definition (multivariate case)",
        "url": "https://en.wikipedia.org/wiki/Polynomial_ring"
      }
    ],
    "tag": "0CES91M79A",
    "uses": [
      "0Q9EZS90GY",
      "0GYGHKX834"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0CES91M79A/",
  "links": [
    {
      "field": "defines",
      "target": "07GNX8GCYP"
    },
    {
      "field": "uses",
      "target": "0Q9EZS90GY"
    },
    {
      "field": "uses",
      "target": "0GYGHKX834"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0CES91M79A.md",
  "msc": [
    "13F20"
  ],
  "name": "Polynomials in several variables",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0CE/0CES91M79A.md",
  "status": null,
  "tag": "0CES91M79A",
  "uri": "m2://0CES91M79A",
  "url": "https://math2.tamnd.com/wiki/polynomials-in-several-variables-definition/",
  "writing": {
    "tag": "0CES91M79A",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}