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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 Let be the algebra of the polynomial functions over V. The dimension of V is any of the following integers. It does not change if K is enlarged, if L is replaced by another algebraically closed extension of K and if I is replaced by another ideal having the same zeros (that is having the same radical). 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. The dimension of V is
- The maximal length of the chains of distinct nonempty (irreducible) subvarieties of V.
- The Krull dimension of the coordinate ring A.
- The maximal Krull dimension of the local rings at the points of V.
This definition shows that the dimension is a local property if is irreducible. If is irreducible, it turns out that all the local rings at closed points have the same Krull dimension (see ).
- If V is a variety, the Krull dimension of the local ring at any point of V
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).
- The number of hyperplanes or hypersurfaces in general position which are needed to have an intersection with 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 A.
This the algebraic translation of the preceding definition.
- The difference between n and the maximal length of the regular sequences contained in I.
This is the algebraic translation of the fact that the intersection of n – d general hypersurfaces is an algebraic set of dimension d.
- The degree of the Hilbert polynomial of A.
- The degree of the denominator of the Hilbert series of A.
- 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 I is a prime ideal (i.e. V is an algebraic variety), the transcendence degree over K of the field of fractions of A.
This allows to prove easily that the dimension is invariant under birational equivalence.
Dimension of a projective algebraic set
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 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). Then each monomial like is replaced by the product of the variables in it: 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.
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:
- The real dimension of is the dimension of its Zariski closure.
- The real dimension of is the maximal integer such that there is a homeomorphism of in .
- The real dimension of is the maximal integer such that there is a projection of over a -dimensional subspace with a non-empty interior.
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 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.
- Chapter 11 of Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8.
- 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.
- Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise (2003), Algorithms in Real Algebraic Geometry (PDF), Algorithms and Computation in Mathematics, 10, Springer-Verlag
- Ivan, Bannwarth; Mohab, Safey El Din (2015), Probabilistic Algorithm for Computing the Dimension of Real Algebraic Sets, Proceedings of the 2015 international symposium on Symbolic and algebraic computation, ACM