Combinatorial Dimension Theory of Algebraic Varieties

How can one compute the dimension of a projective algebraic subset, given by a system of equations? There are different definitions, and their equivalence is part of the classical dimension theory in algebraic geometry. The game is then the following: to make some definition of dimension effective enough to get an algorithm. A first method consists to think algebraically and to use the Hilbert function of the quotient ring. It can be easily computed once a particular set of generators (a so called standard basis (=Gr6bner basis) of the defining ideal) is known. In fact such a basis yields a monomial ideal with the same Hilbert function, hence the same Hilbert polynomial, whose degree is the dimension. So it is the order at,infinity of this numerical function, attained for example through the maximal dimension of a coordinate plane containing no elements of the monomial ideal. Based on this idea there are different algorithms already known in the literature (see Carrel Ferro, 1986; Kandri-Rody, 1985; Kredel & Weispfennig, 1988; Lejeune-Jalabert, 1984-1985). Unfortunately this method needs to construct a whole standard basis from a given set of generators, and leads necessarily to a disastrous upper bound for the worst case complexity, as shown by the explicit examples of Mayr & Meyer (1982), Demazure (1985): there are ideals for which deciding whether a given polynomial belongs to them needs exponential space. As the knowledge of a standard basis easily solves the previous problem, computing such a basis needs also exponential space. In order to bypass that, why not to determine the Hilbert polynomial by interpolation, if we know an upper bound for the regularity of the Hilbert function? Alas, it happens that the same catastrophic behaviour cannot be avoided, as indicated in Giusti (1984). Another way is to think geometrically and to cut the variety by linear subspaces, as proposed by Lazard (1981; 1982). Once again this dimension can be read on a standard