In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a semialgebraic set is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
''S'' of ''R
n'' for some
real closed field
In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.
Def ...
''R'' (for example ''R'' could be the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s) defined by a finite sequence of
polynomial equations
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
(of the form
) and
inequalities
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
(of the form
), or any finite
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of such sets. A semialgebraic function is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
with a semialgebraic
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
. Such sets and functions are mainly studied in
real algebraic geometry In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial ...
which is the appropriate framework for
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
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
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class ...
of a semialgebraic set is again semialgebraic. Finally, and most importantly, the
Tarski–Seidenberg theorem In mathematics, the Tarski–Seidenberg theorem states that a set in (''n'' + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto ''n''-dimensional space, and the resulting set is still defina ...
says that they are also closed under the projection operation: in other words a semialgebraic set projected onto a
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
yields another such (as case of
elimination of quantifiers Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "\exists x such that \ldots" can be viewed as a question "When is there an x such ...
). These properties together mean that semialgebraic sets form an
o-minimal structure In mathematical logic, and more specifically in model theory, an infinite structure (''M'',<,...) which is totally ordered by < is called an o-minimal structure if and only if every 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
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of the semialgebraic set ''S'', it is (locally) a
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...
. 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 In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ : ''U'' → ...
*
Existential theory of the reals In mathematical logic, computational complexity theory, and computer science, the existential theory of the reals is the set of all true sentences of the form
\exists X_1 \cdots \exists X_n \, F(X_1,\dots, X_n),
where the variables X_i are interpre ...
*
Subanalytic set In mathematics, particularly in the subfield of real analytic geometry, a subanalytic set is a set of points (for example in Euclidean space) defined in a way broader than for semianalytic sets (roughly speaking, those satisfying conditions requirin ...
References
*.
*.
*{{citation, first=L., last=van den Dries, title=Tame topology and ''o''-minimal structures, publisher=Cambridge University Press, year=1998, isbn=9780521598385, url=https://books.google.com/books?id=CLnElinpjOgC&q=semialgebraic.
External links
PlanetMath page
Real algebraic geometry