Semialgebraic set

Semialgebraic set

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.

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). These properties together mean that semialgebraic sets form an o-minimal structure on R.

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.

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

*L. van den Dries, "Tame topology and o-minimal structures", Cambridge University Press, 1998.

External links

* [http://planetmath.org/?op=getobj&from=objects&id=8997 PlanetMath page]


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