Semialgebraic set
Template:Short description 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 the main object of study of real algebraic geometry the part of algebraic geometry devoted to points with real coordinates.
Tarski–Seidenberg theorem implies that the semialgebraic sets are exactly those sets that can be defined by a well-formed formula built with quantifiers Template:Tmath, logical connectives Template:Tmath, basic arithmetic operators Template:Tmath, equality and inequality signs Template:Tmath, integers, and variables representing real numbers.
Definition
Let be the field of real numbers, or, more generally, any real closed field.
A basic semialgebraic set in Template:Tmath is the set of the common solutions of a finite number of polynomial equations and inequalities of the form or where Template:Tmath and Template:Tmath are polynomial coefficients
A semialgebraic set in Template:Tmath is a finite union of basic semialgebraic sets.
Tarski–Seidenberg theorem and basic rules of Boolean algebra and algebra imply that semialgebra sets are exactly those subsets of Template:Tmath that can be defined by a well-formed formula built from the quantifiers Template:Tmath, the logical connectives Template:Tmath, equality and inequality signs Template:Tmath, the arithmetic operators Template:Tmath, integers, and the variables Template:Tmath.
So, the definition of semialgebraic sets may be viewed as a kind of normal form for this general class of first-order formulas.
Dimension 1
In Template:Tmath, a semialgebraic set is the union a finite number of intervals whose end points are algebraic numbers; this includes zero-length intervals that are each reduced to a single algebraic number. Conversely, such an union is a always a semialgebraic set.
For proving this we define first some special basic semialgebraic sets. Then we prove that every basic algebraic sset is a disjoint union of such special sets and that this implies the result.
By definition of an algebraic number, an algebraic number Template:Tmath is a root of a square-free polynomial Template:Tmath with integer coefficients and positive leading coefficient (the square-free condition implies that the polynomial changes of sign at each root). Real-root isolation provides algorithms for computing two rational numbers Template:Tmath and Template:Tmath such that Template:Tmath and there is no other root of Template:Tmath in the interval Template:Tmath. So, the singleton is a basic semialgebraic set.
With the same notation, if Template:Tmath is the largest root of Template:Tmath, the open interval Template:Tmath a basic semialgebraic set, since
Similarly, if Template:Tmath is the smallest root, one has the basic semialgebraic set
If Template:Tmath are two consecutive roots of Template:Tmath, real-root isolation provides three rational numbers Template:Tmath, Template:Tmath and Template:Tmath such that Template:Tmath and the interval Template:Tmath does not contain any other root of Template:Tmath. In this case, expresses Template:Tmath as a basic algebraic set.
Now, given a basic semialgebraic set, there are two cases: firstly if its definition contains at least one equation Template:Tmath, the set consists of a subset of the roots of Template:Tmath, and is thus the union of above special basic sets.
If there is no equation in the definition of the basic algebraic set, it has the form Since each Template:Tmath has a constant sign in each open interval delimited by two consecutive roots of the product Template:Tmath, the set Template:Tmath is the union of some ot these intervals, and thus a union of special basic semialgebraic sets.
The final result follows since a finite union of singletons and open intervals is also a disjoint union of points nd intervals.
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.