Login
Login

semialgebraic set

{{short description|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 \mathbb{F} be a real closed field (For example \mathbb{F} could be the field of real numbers \mathbb{R}).

A subset S of \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) \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) \in\mathbb{F}^n \mid P(x_1,...,x_n) > 0\}.

Image:Annulus area.svg

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

  • {{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|isbn=9783662037188|url=https://books.google.com/books?id=GJv6CAAAQBAJ&q=Semialgebraic}}.
  • {{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|s2cid=56006439}}.
  • {{citation|first=L.|last=van den Dries|title=Tame topology and o-minimal structures|publisher=Cambridge University Press|year=1998|isbn=9780521598385|url=https://books.google.com/books?id=CLnElinpjOgC&q=semialgebraic}}.

External links

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

Category:Real algebraic geometry