In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways.
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
be the K-algebra of the polynomial functions over V. The dimension of V is any of the following integers.
The dimension is also independent of the choice of coordinates; in other words it does not change if the xi are replaced by linearly independent linear combinations of them.
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 if
is irreducible, it turns out that all the local rings at points of V have the same Krull dimension (see [1]); thus: This rephrases the previous definition into a more geometric language.
This relates 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, if it is not empty, 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 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).
This definition is not intrinsic as it apply only to algebraic sets that are explicitly embedded in an affine or projective space.
This is the algebraic translation of the fact that the intersection of n – d general hypersurfaces is 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.
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] This allows to prove easily that the dimension is invariant under birational equivalence.
Let V be a projective algebraic set defined as the set of the common zeros of a homogeneous ideal I in a polynomial ring
over a field K, and let 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 A or I appear explicitly in the definition, the value of the dimension must be reduced by one.
Given a system of polynomial equations over an algebraically closed field
, 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).
The real dimension of a set of real points, typically a semialgebraic set, is the dimension of its Zariski closure.