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Inmathematics and specifically inalgebraic geometry, thedimension of analgebraic variety may be defined in various equivalent ways.

Some of these definitions are of geometric nature, while some other are purely algebraic and rely oncommutative algebra. Some are restricted to algebraic varieties while others apply also to anyalgebraic set. Some are intrinsic, as independent of anyembedding of the variety into anaffine orprojective space, while other are related to such an embedding.

LetK be afield, andL ⊇K be analgebraically closed extension.

Anaffine algebraic setV is the set of the commonzeros inLⁿ of the elements of anidealI in apolynomial ring${\displaystyle R=K[x_1,\dots ,x_n].}$ Let${\displaystyle A=R/I}$ be theK-algebra of the polynomial functions overV. The dimension ofV is any of the following integers. It does not change ifK is enlarged, ifL is replaced by another algebraically closed extension ofK and ifI is replaced by another ideal having the same zeros (that is having the sameradical). The dimension is also independent of the choice of coordinates; in other words it does not change if thex_i are replaced bylinearly independentlinear combinations of them.

The dimension ofV is

• The maximal length${\displaystyle d}$ of the chains${\displaystyle V_0\subset V_1\subset \dots \subset V_d}$of distinct nonempty (irreducible) subvarieties ofV.

This definition generalizes a property of the dimension of aEuclidean space or avector space. It is thus probably the definition that gives the easiest intuitive description of the notion.

• TheKrull dimension of thecoordinate ringA.

This is the transcription of the preceding definition in the language ofcommutative algebra, the Krull dimension being the maximal length of the chains${\displaystyle p_0\subset p_1\subset \dots \subset p_d}$ ofprime ideals ofA.

• The maximal Krull dimension of thelocal rings at the points ofV.

This definition shows that the dimension is alocal property if${\displaystyle V}$ is irreducible. If${\displaystyle V}$ is irreducible, it turns out that all the local rings at points ofV have the same Krull dimension (see^[1]); thus:

• IfV is a variety, the Krull dimension of the local ring at any point ofV

This rephrases the previous definition into a more geometric language.

• The maximal dimension of thetangent vector spaces at the nonsingular points ofV.

This relates the dimension of a variety to that of adifferentiable manifold. More precisely, ifV 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.

• IfV is a variety, the dimension of thetangent vector space at any nonsingular point ofV.

This is the algebraic analogue to the fact that a connectedmanifold 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 (seeZariski tangent space).

• The number ofhyperplanes orhypersurfaces ingeneral position which are needed to have an intersection withV 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 aregular sequence in the coordinate ringA.

This the algebraic translation of the preceding definition.

• The difference betweenn and the maximal length of the regular sequences contained inI.

This is the algebraic translation of the fact that the intersection ofn –d general hypersurfaces is an algebraic set of dimensiond.

• The degree of theHilbert polynomial ofA.
• The degree of the denominator of theHilbert series ofA.

This allows, through aGröbner basis computation to compute the dimension of the algebraic set defined by a givensystem 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:^[2]

• The Krull dimension of theStanley–Reisner ring${\displaystyle R/J}$ where${\displaystyle J}$ is theradical of theinitial ideal of${\displaystyle I}$ for any admissiblemonomial ordering (theinitial ideal of${\displaystyle I}$ is the set of all leading monomials of elements of${\displaystyle I}$).
• The dimension of thesimplicial complex defined by this Stanley–Reisner ring.
• IfI is a prime ideal (i.e.V is an algebraic variety), thetranscendence degree overK of thefield of fractions ofA.

This allows to prove easily that the dimension is invariant underbirational equivalence.

LetV be aprojective algebraic set defined as the set of the common zeros of a homogeneous idealI in a polynomial ring${\displaystyle R=K[x_0,x_1,\dots ,x_n]}$ over a fieldK, and letA=R/I be thegraded algebra of the polynomials overV.

All the definitions of the previous section apply, with the change that, whenA orI appear explicitly in the definition, the value of the dimension must be reduced by one. For example, the dimension ofV is one less than the Krull dimension ofA.

Given asystem of polynomial equations over an algebraically closed field${\displaystyle K}$, 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 theHilbert 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${\displaystyle {x_1}^{e_1}\cdots {x_n}^{e_n}}$ is replaced by the product of the variables in it:${\displaystyle x_1^{min(e_1,1)}\cdots x_n^{min(e_n,1)}.}$ Then the dimension is the maximal size of a subsetS of the variables, such that none of these products of variables depends only on the variables inS.

This algorithm is implemented in severalcomputer algebra systems. For example inMaple, this is the functionGroebner[HilbertDimension], and inMacaulay2, this is the functiondim.

Thereal dimension of a set of real points, typically asemialgebraic set, is the dimension of itsZariski closure. For a semialgebraic setS, the real dimension is one of the following equal integers:^[3]

• The real dimension of${\displaystyle S}$ is the dimension of its Zariski closure.
• The real dimension of${\displaystyle S}$ is the maximal integer${\displaystyle d}$ such that there is ahomeomorphism of${\displaystyle [0,1{]}^d}$ in${\displaystyle S}$.
• The real dimension of${\displaystyle S}$ is the maximal integer${\displaystyle d}$ such that there is aprojection of${\displaystyle S}$ over a${\displaystyle d}$-dimensional subspace with a non-emptyinterior.

For an algebraic set defined over thereals (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, thealgebraic surface of equation${\displaystyle x^2+y^2+z^2=0}$ 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 realhypersurface (that is the set of real solutions of a single polynomial equation), there exists a probabilistic algorithm to compute its real dimension.^[4]

• Dimension (vector space)
• Dimension theory (algebra)
• Dimension of a scheme

• Algebraic varieties
• Dimension
• Computer algebra

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