semialgebraic set


Consider the An, defined by real polynomials pj, j=1,,k, =1,,m, and the relationsMathworldPlanetmath ϵj where ϵj is >, =, or <.

A==1m{xnpj(x)ϵj0,j=1,,k}. (1)

Sets of this form are said to be semialgebraic.

Similarly as algebraic subvarietiesMathworldPlanetmath, finite union and intersectionMathworldPlanetmath 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 submanifoldMathworldPlanetmath, 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.


  • 1 Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5–42. 89k:32011
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