Tarski-Seidenberg theorem

Theorem (Tarski-Seidenberg).

The set of semialgebraic sets is closed under projection.

That is, if $A\\subset{\\mathbb{R}}^{n}\\times{\\mathbb{R}}^{m}$ is a semialgebraic set, and if $\\pi$ is the projection onto the first $n$ coordinates, then $\\pi(A)$ is also semialgebraic.

Łojasiewicz generalized this theorem further. For this we need a bit of notation.

Let $U\\subset{\\mathbb{R}}^{n}$ .Suppose $\\mathcal{A}(U)$ is any ring of real valued functions on $U$ .Define $\\mathcal{S}(\\mathcal{A}(U))$ to be the smallestset of subsets of $U$ , which contain the sets $\\{x\\in U\\mid f(x)>0\\}$ for all $f\\in\\mathcal{A}(U)$ ,and is closed under finite union, finite intersection and complement.Let $\\mathcal{A}(U)[t]$ denote the ring of polynomials in $t\\in{\\mathbb{R}}^{m}$ with coefficients in $\\mathcal{A}(U)$ .

Theorem (Tarski-Seidenberg-Łojasiewicz).

Suppose that $V\\subset U\\times{\\mathbb{R}}^{m}\\subset{\\mathbb{R}}^{n+m}$ ,is such that $V\\in\\mathcal{S}(\\mathcal{A}(U)[t])$ .Then the projection of $V$ onto the first $n$ variablesis in $\\mathcal{S}(\\mathcal{A}(U))$ .

References

1 Edward Bierstone and Pierre D. Milman, Semianalytic and subanalyticsets, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5–42.http://www.ams.org/mathscinet-getitem?mr=89k:32011MR 89k:32011

单词	TarskiSeidenbergTheorem
释义	Tarski-Seidenberg theorem Theorem (Tarski-Seidenberg). The set of semialgebraic sets is closed under projection. That is, if $A\\subset{\\mathbb{R}}^{n}\\times{\\mathbb{R}}^{m}$ is a semialgebraic set, and if $\\pi$ is the projection onto the first $n$ coordinates, then $\\pi(A)$ is also semialgebraic. Łojasiewicz generalized this theorem further. For this we need a bit of notation. Let $U\\subset{\\mathbb{R}}^{n}$ .Suppose $\\mathcal{A}(U)$ is any ring of real valued functions on $U$ .Define $\\mathcal{S}(\\mathcal{A}(U))$ to be the smallestset of subsets of $U$ , which contain the sets $\\{x\\in U\\mid f(x)>0\\}$ for all $f\\in\\mathcal{A}(U)$ ,and is closed under finite union, finite intersection and complement.Let $\\mathcal{A}(U)[t]$ denote the ring of polynomials in $t\\in{\\mathbb{R}}^{m}$ with coefficients in $\\mathcal{A}(U)$ . Theorem (Tarski-Seidenberg-Łojasiewicz). Suppose that $V\\subset U\\times{\\mathbb{R}}^{m}\\subset{\\mathbb{R}}^{n+m}$ ,is such that $V\\in\\mathcal{S}(\\mathcal{A}(U)[t])$ .Then the projection of $V$ onto the first $n$ variablesis in $\\mathcal{S}(\\mathcal{A}(U))$ . References 1 Edward Bierstone and Pierre D. Milman, Semianalytic and subanalyticsets, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5–42.http://www.ams.org/mathscinet-getitem?mr=89k:32011MR 89k:32011
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