# Semialgebraic set

In mathematics, a **semialgebraic set** is a subset *S* of *R ^{n}* for some real closed field

*R*(for example

*R*could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form ) and inequalities (of the form ), or any finite union of such sets. A

**semialgebraic function**is a function with semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry which is the appropriate framework for algebraic geometry over the real numbers.

## 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*.

A semialgebraic set (or function) is said to be **defined over a subring** *A* of *R* if there is some description as in the definition, where the polynomials can be chosen to have coefficients in *A*.

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

- Bochnak, J.; Coste, M.; Roy, M.-F. (1998),
*Real algebraic geometry*, Berlin: Springer-Verlag. - Bierstone, Edward; Milman, Pierre D. (1988), "Semianalytic and subanalytic sets",
*Inst. Hautes Études Sci. Publ. Math.*,**67**: 5–42, doi:10.1007/BF02699126, MR 0972342. - van den Dries, L. (1998),
*Tame topology and*o*-minimal structures*, Cambridge University Press.