Semi-algebraic 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 ${\displaystyle P(x_{1},...,x_{n})=0}$) and inequalities (of the form ${\displaystyle Q(x_{1},...,x_{n})>0}$), 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.

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.

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