not empty, is a differentiable manifold that has the same dimension as a variety and as a manifold. If is a variety, the dimension of the tangent vector space at any non singular point of . 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 hyperplanes or hypersurfaces in general position which are needed to have an intersection with 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 . This the algebraic translation of the preceding definition. The difference between and the maximal length of the regular sequences contained in . This is the algebraic translation of the fact that the intersection of general hypersurfaces is an algebraic set of dimension . The degree of the Hilbert polynomial of . The degree of the denominator of the Hilbert series of . This allows, through a Gröbner basis computation to compute the dimension of the algebraic set defined by a given system of polynomial equations. The dimension of the simplicial complex whose Stanley-Reisner ring is where is the radical of any initial ideal of I. Taking initial ideals preserves Hilbert polynomial/series, and taking radicals preserves the dimension. If is a prime ideal (i.e. is an algebraic variety), the transcendence degree over of the field of fractions of . 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 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. For example, the dimension of V is one less than the Krull dimension of A. Computation of the dimension Given a system