{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $R$ be a [[real closed field|DDY3Q3PJ6W]], for example $\\R$.\nA subset of $R^m$ is *semialgebraic* if it is a finite union of sets of the following form.\n$$\\set{x \\in R^m : f(x) = 0, \\ g_1(x) \u003e 0, \\dots, g_k(x) \u003e 0}$$\nHere $k \\ge 0$, and $f, g_1, \\dots, g_k$ are [[polynomials|07GNX8GCYP]] in $m$ variables with coefficients in $R$.\nLet $n \\ge 1$, let $\\pi \\colon R^{n+1} \\to R^n$ be the projection $\\pi(x_1, \\dots, x_{n+1}) = (x_1, \\dots, x_n)$, and let $S \\subseteq R^{n+1}$ be semialgebraic.\nThen $\\pi(S)$ is a semialgebraic subset of $R^n$.\n\n## Remarks\n\nThe theorem is the geometric form of [[quantifier elimination for real closed fields|FSJ9NGRRPA]].\nThe set $\\pi(S)$ is defined by the formula $\\exists x_{n+1} \\, \\varphi(x_1, \\dots, x_{n+1})$, where $\\varphi$ is a quantifier-free formula that defines $S$.\nQuantifier elimination replaces it by a quantifier-free formula with parameters in $R$, and the set defined by such a formula is semialgebraic.\n\nExample: let $S = \\set{(a, x) \\in \\R^2 : x^2 - a = 0}$, which is semialgebraic.\nIts projection to the first coordinate is $\\set{a : a \\ge 0}$, which is semialgebraic as the union of $\\set{a : a = 0}$ and $\\set{a : a \u003e 0}$.\n\nThe analogous statement fails for algebraic sets.\nThe projection of the algebraic set $S$ above is $[0, \\infty)$, which is not the zero set of any family of polynomials in one variable.\nThis is why inequalities are part of the definition.\n\n## Explanations\n\n### L1 human\n\nSemialgebraic sets are the sets cut out by polynomial equations and inequalities, such as discs, half-planes and parabolic regions.\nThe theorem says that the shadow of such a set on a lower-dimensional space is again of this kind.\nSo the class of semialgebraic sets is stable under the operations that geometric constructions use.\n\n### L2 human\n\nBy induction on $n$, the image of a semialgebraic set under any polynomial map is semialgebraic, and so are closures, interiors and sets defined with quantifiers over $R$.\nThis makes semialgebraic sets the natural setting of real algebraic geometry, in contrast with real algebraic sets, which lack this closure property.\nIt also shows that every subset of $R$ definable with $+$, $\\cdot$ and $\u003c$ is semialgebraic, hence a finite union of points and open intervals.\nThis is the starting point of o-minimality.\nAlgorithms for quantifier elimination over $\\R$, such as cylindrical algebraic decomposition, make the projection effective.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "about": [
      "DDY3Q3PJ6W"
    ],
    "kind": "statement",
    "msc": [
      "14P10",
      "03C10",
      "12D15"
    ],
    "name": "Tarski–Seidenberg theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q16705451",
      "wikipedia": "Tarski–Seidenberg theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Tarski–Seidenberg_theorem"
      }
    ],
    "tag": "3W5JGDZAN0",
    "uses": [
      "DDY3Q3PJ6W",
      "FSJ9NGRRPA",
      "07GNX8GCYP"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/3W5JGDZAN0/",
  "links": [
    {
      "field": "about",
      "target": "DDY3Q3PJ6W"
    },
    {
      "field": "uses",
      "target": "DDY3Q3PJ6W"
    },
    {
      "field": "uses",
      "target": "FSJ9NGRRPA"
    },
    {
      "field": "uses",
      "target": "07GNX8GCYP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/3W5JGDZAN0.md",
  "msc": [
    "14P10",
    "03C10",
    "12D15"
  ],
  "name": "Tarski–Seidenberg theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3W5/3W5JGDZAN0.md",
  "status": null,
  "tag": "3W5JGDZAN0",
  "uri": "m2://3W5JGDZAN0",
  "url": "https://math2.tamnd.com/wiki/tarski-seidenberg-theorem/",
  "writing": {
    "tag": "3W5JGDZAN0",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}