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((short description|Measure of a mathematical object studied in the field of algebraic geometry))
((refimprove|date=April 2016))
In [[mathematics]] and specifically in [[algebraic geometry]], the '''dimension''' of an [[algebraic variety]] may be defined in various equivalent ways.

Some of these definitions are of geometric nature, while some other are purely algebraic and rely on [[commutative algebra]]. Some are restricted to algebraic varieties while others apply also to any [[algebraic set]]. Some are intrinsic, as independent of any [[embedding]] of the variety into an [[affine space|affine]] or [[projective space]], while other are related to such an embedding.

== Dimension of an affine algebraic set ==

Let ((math|''K'')) be a [[field (mathematics)|field]], and ((math|''L'' ⊇ ''K'')) be an [[algebraically closed extension]]. 

An [[affine algebraic set]] ((math|''V'')) is the set of the common [[zero of a function|zeros]] in ((math|''L''<sup>''n''</sup>)) of the elements of an [[ideal of a ring|ideal]] ((math|''I'')) in a [[polynomial ring]] <math>R=K[x_1, \ldots, x_n].</math> Let <math>A=R/I</math> be the [[algebra over a field|''K''-algebra]] of the polynomial functions over ((math|''V'')). The dimension of ((math|''V'')) is any of the following integers. It does not change if ((math|''K'')) is enlarged, if ((math|''L'')) is replaced by another algebraically closed extension of ((math|''K'')) and if ((math|''I'')) is replaced by another ideal having the same zeros (that is having the same [[radical of an ideal|radical]]). The dimension is also independent of the choice of coordinates; in other words it does not change if the ((math|''x''<sub>''i''</sub>)) are replaced by [[linearly independent]] [[linear combination]]s of them. 

The dimension of ((math|''V'')) is
* ''The maximal length'' <math>d</math> of the chains <math>V_0\subset V_1\subset \ldots \subset V_d</math> ''of distinct nonempty (irreducible) subvarieties of ((math|V)).''
This definition generalizes a property of the dimension of a [[Euclidean space]] or a [[vector space]]. It is thus probably the definition that gives the easiest intuitive description of the notion. 
* ''The [[Krull dimension]] of the [[coordinate ring]] ((mvar|A)).''
This is the transcription of the preceding definition in the language of [[commutative algebra]], the Krull dimension being the maximal length of the chains <math>p_0\subset p_1\subset \ldots \subset p_d</math> of [[prime ideal]]s of ((math|''A'')).
* ''The maximal Krull dimension of the [[local ring]]s at the points of ((mvar|V))''.

This definition shows that the dimension is a ''local property if <math>V</math>is irreducible.'' If <math>V</math>is irreducible, it turns out that all the local rings at points of ((mvar|V)) have the same Krull dimension (see <ref>Chapter 11 of Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ((ISBN|978-0-201-40751-8)).</ref>); thus:

* ''If ((mvar|V)) is a variety, the Krull dimension of the local ring at any point of ((mvar|V))''

This rephrases the previous definition into a more geometric language.
* ''The maximal dimension of the [[tangent space|tangent vector space]]s at the non [[singular point of an algebraic variety|singular point]]s of ((mvar|V))''.
This relates the dimension of a variety to that of a [[differentiable manifold]]. More precisely, if ((math|''V'')) if defined over the reals, then the set of its real regular points, if it is not empty, is a differentiable manifold that has the same dimension as a variety and as a manifold.
* ''If ((mvar|V)) is a variety, the dimension of the [[tangent space|tangent vector space]] at any non [[singular point of an algebraic variety|singular point]] of ((mvar|V))''.
This is the algebraic analogue to the fact that a connected [[manifold]] has a constant dimension. This can also be deduced from the result stated below the third definition, and the fact that the dimension of the tangent space is equal to the Krull dimension at any non-singular point (see [[Zariski tangent space]]).
* ''The number of [[hyperplane]]s or [[hypersurface]]s in [[general position]] which are needed to have an intersection with ((mvar|V)) which is reduced to a nonzero finite number of points.''
This definition is not intrinsic as it apply only to algebraic sets that are explicitly embedded in an affine or projective space.
* ''The maximal length of a [[regular sequence]] in the coordinate ring ((mvar|A))''.
This the algebraic translation of the preceding definition.
* ''The difference between ((mvar|n)) and the maximal length of the regular sequences contained in ((mvar|I))''.
This is the algebraic translation of the fact that the intersection of ((math|''n'' – ''d'')) general hypersurfaces is an algebraic set of dimension ((math|''d'')). 
* ''The degree of the [[Hilbert polynomial]] of ((mvar|A))''.
* ''The degree of the denominator of the [[Hilbert series]] of ((mvar|A))''.
This allows, through a [[Gröbner basis]] computation to compute the dimension of the algebraic set defined by a given [[system of polynomial equations]]. Moreover, the dimension is not changed if the polynomials of the Gröbner basis are replaced with their leading monomials, and if these leading monomials are replaced with their radical (monomials obtained by removing exponents). So:<ref> Cox, David A.; Little, John; O'Shea, Donal Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. Fourth edition. Undergraduate Texts in Mathematics. Springer, Cham, 2015.</ref>
* ''The Krull dimension of the [[Stanley–Reisner ring]] <math>R/J</math> where <math>J</math> is the [[Radical of an ideal|radical]] of the ''initial ideal'' of <math>I</math> for any admissible [[monomial ordering]] (the ''initial ideal'' of <math>I</math> is the set of all leading monomials of elements of <math>I</math>).''
* ''The dimension of the [[simplicial complex]] defined by this Stanley–Reisner ring.
* ''If ((mvar|I)) is a prime ideal (i.e. ((mvar|V)) is an algebraic variety), the [[transcendence degree]] over ((mvar|K)) of the [[field of fractions]] of ((mvar|A))''.
This allows to prove easily that the dimension is invariant under [[birational equivalence]].

== Dimension of a projective algebraic set ==

Let ((math|''V'')) be a [[projective algebraic set]] defined as the set of the common zeros of a homogeneous ideal ((math|''I'')) in a polynomial ring <math>R=K[x_0, x_1, \ldots, x_n]</math> over a field ((math|''K'')), and let ((math|1=''A''=''R''/''I'')) be the [[graded algebra]] of the polynomials over ''V''.

All the definitions of the previous section apply, with the change that, when ((math|''A'')) or ((math|''I'')) appear explicitly in the definition, the value of the dimension must be reduced by one. For example, the dimension of ((math|''V'')) is one less than the Krull dimension of ((math|''A'')).

== Computation of the dimension ==

Given a [[system of polynomial equations]] over an algebraically closed field <math>K</math>, it may be difficult to compute the dimension of the algebraic set that it defines.

Without further information on the system, there is only one practical method, which consists of computing a Gröbner basis and deducing the degree of the denominator of the [[Hilbert series]] of the ideal generated by the equations.

The second step, which is usually the fastest, may be accelerated in the following way: Firstly, the Gröbner basis is replaced by the list of its leading monomials (this is already done for the computation of the Hilbert series). Then each monomial like <math>{x_1}^{e_1}\cdots {x_n}^{e_n}</math> is replaced by the product of the variables in it: <math>x_1^{\min (e_1,1)}\cdots x_n^{\min(e_n,1)}.</math> Then the dimension is the maximal size of a subset ''S'' of the variables, such that none of these products of variables depends only on the variables in ''S''.

This algorithm is implemented in several [[computer algebra system]]s. For example in [[Maple (software)|Maple]], this is the function ''Groebner[HilbertDimension],'' and in [[Macaulay2]], this is the function ''dim''.

== Real dimension ==
((see also|Complex dimension))
The ''real dimension'' of a set of real points, typically a [[semialgebraic set]], is the dimension of its [[Zariski closure]]. For a semialgebraic set ((mvar|S)), the real dimension is one of the following equal integers:<ref>((citation
|first1=Saugata|last1=Basu
|first2=Richard|last2=Pollack
|first3=Marie-Françoise|last3=Roy
|title= Algorithms in Real Algebraic Geometry
|publisher=  Springer-Verlag
|series = Algorithms and Computation in Mathematics
|volume = 10
|year=2003
|url=https://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted3.pdf))</ref>
* The real dimension of <math>S</math> is the dimension of its Zariski closure.
* The real dimension of <math>S</math> is the maximal integer <math>d</math> such that there is a [[homeomorphism]] of <math>[0,1]^d</math> in <math>S</math>.
* The real dimension of <math>S</math> is the maximal integer <math>d</math> such that there is a [[Affine space#Projection|projection]] of <math>S</math> over a <math>d</math>-dimensional subspace with a non-empty [[interior (topology)|interior]].

For an algebraic set defined over the [[real number|reals]] (that is defined by polynomials with real coefficients), it may occur that the real dimension of the set of its real points is smaller than its dimension as a semi algebraic set. For example, the [[algebraic surface]] of equation <math>x^2+y^2+z^2=0</math> is an algebraic variety of dimension two, which has only one real point (0, 0, 0), and thus has the real dimension zero.

The real dimension is more difficult to compute than the algebraic dimension.
For the case of a real [[hypersurface]] (that is the set of real solutions of a single polynomial equation), there exists a probabilistic algorithm to compute its real dimension.<ref>((citation
|first1=Bannwarth|last1=Ivan
|first2=Safey El Din|last2=Mohab
|title= Probabilistic Algorithm for Computing the Dimension of Real Algebraic Sets
|series=Proceedings of the 2015 international symposium on Symbolic and algebraic computation
|publisher=ACM
|year=2015
|url=https://hal.archives-ouvertes.fr/hal-01152751/document))
</ref>

== See also ==
*[[Dimension (vector space)]]
*[[Dimension theory (algebra)]]
*[[Dimension of a scheme]]

==References==
((reflist))

((Dimension topics))

[[Category:Algebraic varieties]]
[[Category:Dimension]]
[[Category:Computer algebra]]

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