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 $$\mathbb{F}$$ 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 $$\{(x_1,...,x_n) \in\mathbb{F}^n \mid P(x_1,...,x_n) > 0\}.$$

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.