# Tarski-Seidenberg theorem

###### Theorem (Tarski-Seidenberg).

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 TarskiSeidenbergTheorem 2013-03-22 16:46:13 2013-03-22 16:46:13 jirka (4157) jirka (4157) 5 jirka (4157) Theorem msc 14P15 msc 14P10 SemialgebraicSet SubanalyticSet Tarski-Seidenberg-Łojasiewicz theorem