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 smallest set 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 $n$ variables is in $\mathcal{S}(\mathcal{A}(U))$ .

References

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

Title	Tarski-Seidenberg theorem
Canonical name	TarskiSeidenbergTheorem
Date of creation	2013-03-22 16:46:13
Last modified on	2013-03-22 16:46:13
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	5
Author	jirka (4157)
Entry type	Theorem
Classification	msc 14P15
Classification	msc 14P10
Related topic	SemialgebraicSet
Related topic	SubanalyticSet
Defines	Tarski-Seidenberg-Łojasiewicz theorem