Semialgebraic set
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In mathematics, a semialgebraic set is a subset S of real n-dimensional space defined by a finite sequence of polynomial equations and inequalities; or any finite union of such sets. Such sets are studied as an extension of real algebraic geometry, in which only equations would be used, and in mathematical logic.
[edit] Properties
Similarly to algebraic subvarieties, finite unions and intersections of semialgebraic sets are still semialgebraic sets. 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 the projection operation: in other words a semialgebraic set projected onto a linear subspace yields another such (as case of elimination of quantifiers).
On a dense open subset of the semialgebraic set S, it is (locally) a submanifold. One can define the dimension of S to be the largest dimension at points at which it is a submanifold. It is not hard to see that a semialgebraic set lies inside an algebraic subvariety of the same dimension.
[edit] References
- J. Bochnak, M. Coste, and M.-F. Roy, Géométrie algébrique réelle, Springer-Verlag, Berlin, 1987.
- Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5-42. MR 89k:32011
