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 ${\displaystyle \mathbb {F} }$  be a real closed field (For example ${\displaystyle \mathbb {F} }$  could be the field of real numbers ${\displaystyle \mathbb {R} }$ ). A subset ${\displaystyle S}$  of ${\displaystyle \mathbb {F} ^{n}}$  is a semialgebraic set if it is a finite union of sets defined by polynomial equalities of the form ${\displaystyle \{(x_{1},...,x_{n})\in \mathbb {F} ^{n}\mid P(x_{1},...,x_{n})=0\}}$  and of sets defined by polynomial inequalities of the form ${\displaystyle \{(x_{1},...,x_{n})\in \mathbb {F} ^{n}\mid P(x_{1},...,x_{n})>0\}.}$ 

 

The annulus (shown in green) is a semialgebraic set in the plane.

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

• Łojasiewicz inequality
• Existential theory of the reals
• Subanalytic set
• Piecewise algebraic space