semialgebraic set

Definition.

Consider the $A\subset {ℝ}^n$, defined by real 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 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.

Title	semialgebraic set
Canonical name	SemialgebraicSet
Date of creation	2013-03-22 16:46:10
Last modified on	2013-03-22 16:46:10
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	5
Author	jirka (4157)
Entry type	Definition
Classification	msc 14P10
Related topic	TarskiSeidenbergTheorem
Related topic	SubanalyticSet
Defines	semialgebraic
Defines	dimension of a semialgebraic set