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# 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. MR 89k:32011

## Mathematics Subject Classification

14P15*no label found*14P10

*no label found*

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