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. This algorithm is implemented in several computer algebra systems. For example in Maple, this is the function Groebner[HilbertDimension], and in Macaulay2, this is the function dim. 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 , 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. See also Dimension theory (algebra) Dimension of a scheme References Algebraic varieties Dimension Computer algebra