This software contains several procedures that allow one to do algebraic geometry calculations using Mathematica. The first step in essentially any calculation is to calculate a Groebner basis. The mathematica command for this is (case sensitive): GroebnerBasis[H,V] Here H is an array of polynomials (say H = {x^2 + y^2, x y}--note the brackets are curly) the V is the array of variables here V = {x,y}. The output is an array of the elements of a Groebner basis relative to Lex order. Later versions will allow more arbitrary monomial orders. Once you have the Groebner basis you can convert to a monomial basis by using the command MkLeads[L,V] Here L = GroebnerBasis[H,V] note that is V the same as in Groebner[H,V] (in the example {x,y}). Now if you want to calculate the dimension of the quotient of the polynomials in the variables V by the ideal one can call Dim[B,n] where n is the number of variables and B = MkLeads[L,V] If you want the hilbert series then you call Hilb[B,n] There are other functions included: Monomials[f,V] This gives a list of all of the monomials of f with coefficients relative to Lex. e.g. Monomials[x^2 y x + 3 x y^2 z + 7 y^2 z^2,{x,y,z}] = {{{2,1,1},1},{{1,2,1},3},{{0,2,2},7}} LeadMon[f,V] This gives the leading monomial relative to Lex. LeadCoeff[f,V] This gives the leading coefficient relative to Lex. Exercises. 1. Prove that the code Dim[B,n] actually calculates the dimension. 2. Let X, Y be 3 x 3 matrices in indeterminates x[1],...,x[9] and y[1],...,y[9] then f_{i,j} be the i,j entry of X.Y - Y.X. Show that the quotent ring of F[x,y] by the ideal is Cohen-Macaulay.