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 ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset of ...
''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.
D ...
''R'' (for example ''R'' 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s) defined by a finite sequence of
polynomial equations (of the form
) and
inequalities (of the form
), or any finite
union of such 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
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
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
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 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 defi ...
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, l ...
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 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
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 ...
. 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
*
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 requir ...
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