Semialgebraic set

Subset of n-space defined by a finite sequence of polynomial equations and inequalities

In mathematics, a basic semialgebraic set is a set defined by polynomial equalities and polynomial inequalities, and a semialgebraic set is a finite union of basic semialgebraic sets. A semialgebraic function is a function with a 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.

Definition

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

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 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.

See also

References

  • 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.

External links

  • PlanetMath page