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semialgebraic set
In mathematics, a semialgebraic set is a subset ''S'' of ''Rn'' for some real closed field ''R'' (for example ''R'' could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form ) and inequalities (of the form ), or any finite union of such sets. A semialgebraic function is a function with 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.
==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 such (as case of elimination of quantifiers). 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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