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Homesemialgebraic set

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# semialgebraic set

###### Definition.

Consider the $A\subset{\mathbb{R}}^{n}$, defined by real polynomials $p_{{j\ell}}$, $j=1,\ldots,k$, $\ell=1,\ldots,m$, and the relations $\epsilon_{{j\ell}}$ where $\epsilon_{{j\ell}}$ is $>$, $=$, or $<$.

$A=\bigcup_{{\ell=1}}^{m}\{x\in{\mathbb{R}}^{n}\mid p_{{j\ell}}(x)~{}\epsilon_{% {j\ell}}~{}0,\ j=1,\ldots,k\}.$ | (1) |

Sets of this form are said to be semialgebraic.

Similarly as algebraic subvarieties, finite union and intersection of semialgebraic sets is still a semialgebraic set. Furthermore, unlike subvarieties, the complement of a semialgebraic set is again semialgebraic. Finally, and most importantly, the Tarski-Seidenberg theorem says that they are also closed under projection.

On a dense open subset of $A$, $A$ is (locally) a submanifold, and hence we can easily define the dimension of $A$ to be the largest dimension at points at which $A$ is a submanifold. It is not hard to see that a semialgebraic set lies inside an algebraic subvariety of the same dimension.

# 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

14P10*no label found*

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