Semialgebraic set

Let ${\displaystyle \mathbb {F} }$ be a real closed field. (For example ${\displaystyle \mathbb {F} }$ could be the field of real numbers ${\displaystyle \mathbb {R} }$.) A subset ${\displaystyle S}$ of ${\displaystyle \mathbb {F} ^{n}}$ is a semialgebraic set if it is a finite union of sets defined by polynomial equalities of the form ${\displaystyle \{(x_{1},...,x_{n})\in \mathbb {F} ^{n}\mid P(x_{1},...,x_{n})=0\}}$ and of sets defined by polynomial inequalities of the form ${\displaystyle \{(x_{1},...,x_{n})\in \mathbb {F} ^{n}\mid Q(x_{1},...,x_{n})>0\}.}$

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 semialgebraic set (as is the case for quantifier elimination). 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.

• Łojasiewicz inequality
• Existential theory of the reals
• Subanalytic set
• piecewise algebraic space

• Bochnak, J.; Coste, M.; Roy, M.-F. (1998), Real algebraic geometry, Berlin: Springer-Verlag, ISBN 9783662037188.
• 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, S2CID 56006439.
• van den Dries, L. (1998), Tame topology and o-minimal structures, Cambridge University Press, ISBN 9780521598385.

• PlanetMath page