In this chapter we describe various commands which operate on standard bases. These break up naturally into five distinct groups: elimination of variables, ideal (or submodule) membership, manipulations where some of the variables are treated as constants, numeric information, and saturation. At the end of this chapter we describe the commands for displaying information about finite free resolutions.
Many of the commands in this chapter simply choose an appropriate subset of either the standard basis or the leading terms of the standard basis. Others, such as hilb and reduce involve computation. In fact, the computation of the Hilbert function often takes more time than computing the standard basis!
Most of the commands of this chapter assume that their argument is a matrix which has previously had a standard basis constructed for the submodule generated by its columns. If the matrix given is not a standard basis, a warning message is given, and the columns of the matrix are used as if they formed a standard basis. This is quite useful in many cases, if you happen to know that the columns of the matrix already form a standard basis. However, one must exert some care! If these columns do not form a standard basis, the resulting answers might not answer the question which you intended to ask.
For example, if I is a 1 by n matrix whose entries generate an ideal I, then ``codim I'' will display a warning message, and also display a number which it claims is the codimension of this ideal. Most likely, this number is wrong: it is the codimension of the monomial ideal generated by the lead terms of the n generators of I. Usually, this is not the correct answer.