{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $f_1, \\dots, f_m, g \\in \\R[x_1, \\dots, x_n]$ be real [[polynomials|07GNX8GCYP]], and let\n$$K = \\set{x \\in \\R^n : f_1(x) \\ge 0, \\dots, f_m(x) \\ge 0}$$\nbe the basic closed [[semialgebraic set|BX5WSK8ZCT]] they define.\nLet $T$ be the preordering generated by $f_1, \\dots, f_m$.\nIt is the set of all sums $\\sum_{e \\in \\set{0,1}^m} \\sigma_e f_1^{e_1} \\cdots f_m^{e_m}$ in which each $\\sigma_e$ is a sum of squares of polynomials.\nThen:\n\n1. $K = \\emptyset$ if and only if $-1 \\in T$.\n2. $g \u003e 0$ on $K$ if and only if there are $p, q \\in T$ with $pg = 1 + q$.\n3. $g \\ge 0$ on $K$ if and only if there are $p, q \\in T$ and an integer $k \\ge 0$ with $pg = g^{2k} + q$.\n4. $g = 0$ on $K$ if and only if there is an integer $k \\ge 0$ with $-g^{2k} \\in T$.\n\n## Remarks\n\nThe result is due to Krivine (1964) and Stengle (1974).\nThe \"if\" directions are direct: for instance, if $pg = 1 + q$ and $x \\in K$, then $p(x) \\ge 0$ and $q(x) \\ge 0$, so $p(x) g(x) \\ge 1$ and $g(x) \u003e 0$.\nThe \"only if\" directions are the substance.\nThey follow from the theory of orderings of rings together with the Tarski–Seidenberg transfer principle.\nThat principle says that a first-order statement about polynomials holds over $\\R$ if and only if it holds over every real closed field.\nExample: $K = \\set{x \\in \\R : -x^2 - 1 \\ge 0}$ is empty, and the certificate is $-1 = (-x^2 - 1) + x^2 \\in T$.\nThe theorem needs the real field: over $\\C$ sign conditions make no sense, and its analogue for equations is [[Hilbert's Nullstellensatz|D85VZWS9QK]].\nThe denominators $p$ cannot be dropped in general, which is why stronger forms, such as Schmüdgen's theorem that $g \\in T$ when $K$ is compact and $g \u003e 0$ on $K$, need extra hypotheses.\n\n## Explanations\n\n### L1 human\n\nHow can one prove that a polynomial is positive on a region defined by polynomial inequalities?\nThe Positivstellensatz says that there is always an algebraic proof: an identity built from squares and from the defining inequalities.\nSuch an identity can be checked by expanding, without solving anything.\n\n### L2 human\n\nThe theorem is the real counterpart of Hilbert's Nullstellensatz, with sums of squares playing the role of the ideal.\nIt contains Artin's solution of [[Hilbert's seventeenth problem|J45F4F13VB]] as the case $m = 0$ of part 3.\nBounded-degree versions of these certificates can be searched for by semidefinite programming, which is the basis of sum-of-squares methods in polynomial optimization.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "BX5WSK8ZCT"
    ],
    "kind": "statement",
    "msc": [
      "14P10",
      "12D15",
      "13J30"
    ],
    "name": "Positivstellensatz",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Positive_polynomial"
      }
    ],
    "tag": "FN734RQYM6",
    "uses": [
      "BX5WSK8ZCT",
      "07GNX8GCYP",
      "08J2M1JYAG",
      "D85VZWS9QK"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/FN734RQYM6/",
  "links": [
    {
      "field": "about",
      "target": "BX5WSK8ZCT"
    },
    {
      "field": "uses",
      "target": "BX5WSK8ZCT"
    },
    {
      "field": "uses",
      "target": "07GNX8GCYP"
    },
    {
      "field": "uses",
      "target": "08J2M1JYAG"
    },
    {
      "field": "uses",
      "target": "D85VZWS9QK"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/FN734RQYM6.md",
  "msc": [
    "14P10",
    "12D15",
    "13J30"
  ],
  "name": "Positivstellensatz",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FN7/FN734RQYM6.md",
  "status": null,
  "tag": "FN734RQYM6",
  "uri": "m2://FN734RQYM6",
  "url": "https://tamnd.github.io/math2/wiki/positivstellensatz/",
  "writing": {
    "tag": "FN734RQYM6",
    "kind": "statement",
    "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
      }
    ]
  }
}