semialgebraic set

Definition.

Consider the $A\subset {ℝ}^n$,defined byreal polynomials $p_{j\mathrm{\ell }}$, $j=1,\mathrm{\dots },k$, $\mathrm{\ell }=1,\mathrm{\dots },m$,and the relations${ϵ}_{j\mathrm{\ell }}$ where ${ϵ}_{j\mathrm{\ell }}$ is $>$, $=$, or .

$$A=\bigcup _{\mathrm{\ell }=1}^m\{x\in {ℝ}^n\mid p_{j\mathrm{\ell }}(x){ϵ}_{j\mathrm{\ell }}0,j=1,\mathrm{\dots },k\}.$$(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 easilydefine 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 setlies inside an algebraic subvariety of the same dimension.