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In [[mathematics]], a '''semialgebraic set''' is a subset ''S'' of ''R<sup>n</sup>'' 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 <math>P(x_1,...,x_n)=0</math>) and inequalities (of the form <math>Q(x_1,...,x_n) > 0</math>), or any [[finite union]] of such sets. A '''semialgebraic function''' is a function with semialgebraic [[Graph of a function|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 subvariety|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.

==See also==
*[[Existential theory of the reals]]
*[[Subanalytic set]]

==References==
*{{citation|first1=J.|last1=Bochnak|first2=M.|last2=Coste|first3=M.-F.|last3=Roy|title=Real algebraic geometry|publisher=Springer-Verlag|location=Berlin|year=1998}}.
*{{citation|first1=Edward|last1=Bierstone|first2=Pierre D.|last2=Milman|title=Semianalytic and subanalytic sets|journal=Inst. Hautes Études Sci. Publ. Math.|year=1988|volume=67|pages=5–42|mr=972342|url=http://www.numdam.org/item?id=PMIHES_1988__67__5_0|doi=10.1007/BF02699126}}.
*{{citation|first=L.|last=van den Dries|title=Tame topology and ''o''-minimal structures|publisher=Cambridge University Press|year=1998}}.

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

[[Category:Real algebraic geometry]]