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]


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Semialgebraic space — In mathematics, especially in real algebraic geometry, a semialgebraic space is a space which is locally isomorphic to a semialgebraic set.DefinitionLet U be an open subset of R n for some n . A semialgebraic function on U is defined to be a… …   Wikipedia

  • Tarski–Seidenberg theorem — In mathematics, the Tarski–Seidenberg theorem states that a set in n +1 dimensional space defined by polynomial identities and inequalities can be projected down onto n dimensional space, and the resulting set is still definable in terms of… …   Wikipedia

  • Sheaf (mathematics) — This article is about sheaves on topological spaces. For sheaves on a site see Grothendieck topology and Topos. In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.… …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • Stengle's Positivstellensatz — In mathematics, Stengle s Positivstellensatz characterizes polynomials which are positive on a given semialgebraic set over the real numbers, or more generally, any real closed field. It can be thought of as an ordered analogue of Hilbert s… …   Wikipedia

  • o-minimal theory — In mathematical logic, and more specifically in model theory, an infinite structure (M,<,...) which is totally ordered by < is called an o minimal structure if and only if every definable subset X ⊂ M (with parameters taken from… …   Wikipedia

  • O-minimal theory — In mathematical logic, and more specifically in model theory, a totally ordered structure ( M , lt;,...) is o minimal if and only if every definable set X sub; M (with parameters) can be realized as a finite union of intervals and points.A theory …   Wikipedia

  • Whitney conditions — In differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965. A finite filtration by closed subsets F i of a smooth manifold such that the… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”