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. http://www.ams.org/mathscinet-getitem?mr=89k:32011MR 89k:32011
Title semialgebraic set SemialgebraicSet 2013-03-22 16:46:10 2013-03-22 16:46:10 jirka (4157) jirka (4157) 5 jirka (4157) Definition msc 14P10 TarskiSeidenbergTheorem SubanalyticSet semialgebraic dimension of a semialgebraic set