Semialgebraic set
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 [math]\displaystyle{ \mathbb{F} }[/math] be a real closed field. (For example [math]\displaystyle{ \mathbb{F} }[/math] could be the field of real numbers [math]\displaystyle{ \mathbb{R} }[/math].) A subset [math]\displaystyle{ S }[/math] of [math]\displaystyle{ \mathbb{F}^n }[/math] is a semialgebraic set if it is a finite union of sets defined by polynomial equalities of the form [math]\displaystyle{ \{(x_1,...,x_n) \in \mathbb{F}^n \mid P(x_1,...,x_n) = 0\} }[/math] and of sets defined by polynomial inequalities of the form [math]\displaystyle{ \{(x_1,...,x_n) \in\mathbb{F}^n \mid P(x_1,...,x_n) \gt 0\}. }[/math]
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, https://books.google.com/books?id=GJv6CAAAQBAJ&q=Semialgebraic.
- Bierstone, Edward; Milman, Pierre D. (1988), "Semianalytic and subanalytic sets", Inst. Hautes Études Sci. Publ. Math. 67: 5–42, doi:10.1007/BF02699126, http://www.numdam.org/item?id=PMIHES_1988__67__5_0.
- van den Dries, L. (1998), Tame topology and o-minimal structures, Cambridge University Press, ISBN 9780521598385, https://books.google.com/books?id=CLnElinpjOgC&q=semialgebraic.
External links
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