In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a basic semialgebraic set is a set defined by polynomial equalities and polynomial inequalities, and a semialgebraic set is a
finite
Finite may refer to:
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect
* "Finite", a song by Sara Gr ...
union of basic semialgebraic sets. A semialgebraic function is a
function 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 discret ...
. Such sets and functions are mainly studied in
real algebraic geometry which is the appropriate framework for
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
over the real numbers.
Definition
Let
be a
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 ...
(For example
could be the
field 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s
).
A
subset
In mathematics, a 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 a ...
of
is a ''semialgebraic set'' if it is a finite union of sets defined by
polynomial equalities of the form
and of sets defined by polynomial
inequalities of the form
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
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping'');
* linearity of a ''polynomial''.
An example of a li ...
yields another semialgebraic set (as is the case for
quantifier elimination
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 ..." can be viewed as a question "When is there an x such ...
). 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
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
open subset
In mathematics, an open set is a generalization of an open interval in the real line.
In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point in it, contains all points of the met ...
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 S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...
. 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
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