In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways.
Contents
- Dimension of an affine algebraic set
- Dimension of a projective algebraic set
- Computation of the dimension
- Real dimension
- References
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 or projective space, while other are related to such an embedding.
Dimension of an affine algebraic set
Let K be a field, and L ⊇ K be an algebraically closed extension. An affine algebraic set V is the set of the common zeros in Ln of the elements of an ideal I in a polynomial ring
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.
This is the transcription of the preceding definition in the language of commutative algebra, the Krull dimension being the maximal length of the chains
This definition shows that the dimension is a local property.
This shows that the dimension is constant on a variety
This relies the dimension of a variety to that of a differentiable manifold. More precisely, if V if defined over the reals, then the set of its real regular points is a differentiable manifold that has the same dimension as a variety and as a manifold.
This is the algebraic analogue to the fact that a connected manifold has a constant dimension.
This definition is not intrinsic as it apply only to algebraic sets that are explicitly embedded in an affine or projective space.
This the algebraic translation of the preceding definition.
This is the algebraic translation of the fact that the intersection of n – d hypersurfaces is, in general, an algebraic set of dimension d.
This allows, through a Gröbner basis computation to compute the dimension of the algebraic set defined by a given system of polynomial equations
This allows to prove easily that the dimension is invariant under birational equivalence.
Dimension of a projective algebraic set
Let V be a projective algebraic set defined as the set of the common zeros of a homogeneous ideal I in a polynomial ring
All the definitions of the previous section apply, with the change that, when A or I appear explicitly in the definition, the value of the dimension must be reduced by one. For example, the dimension of V is one less than the Krull dimension of A.
Computation of the dimension
Given a system of polynomial equations, 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
This algorithm is implemented in several computer algebra systems. For example in Maple, this is the function Groebner[HilbertDimension].
Real 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 S, the real dimension is one of the following equal integers:
For an algebraic set defined over the 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
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.
For the case of an arbitrary system of polynomial equations and inequalities, computing a triangular decomposition of this system into so-called regular semi-algebraic systems yields the dimension of the solution set of this system. The command RealTriangularize of the RegularChains library implements such decomposition.