Semialgebraic set

(Redirected from Semialgebraic sets)

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

edit

Let   be a real closed field (For example   could be the field of real numbers  ). A subset   of   is a semialgebraic set if it is a finite union of sets defined by polynomial equalities of the form   and of sets defined by polynomial inequalities of the form  

Properties

edit

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

edit

References

edit
  • 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.
edit