16.2 Elimination

Since we will not always let the GBA algorithm run until it finds a Gröbner Basis, we will often be dealing with sets which are not Gröbner Basis, but rather an intermediate result. We call such sets of relations partial GB’s.

Commutative Gröbner Basis Algorithms can be used to systematically eliminate variables from a collection (e.g., {pj(x1,,xn) = 0 : 1 j k1}) of polynomial equations so as to put it in triangular form. One specifies an order on the variables (x1 < x2 < x3 < < xn ) 3 which corresponds to ones priorities in eliminating them. Here a GBA will try hardest to eliminate xn and try the least to eliminate x1. The output from it is a list of equations in a “canonical form” which is triangular: 4

         q1(x1) = 0                                (16.1)

      q2(x1,x2) = 0                                (16.2)
      q3(x1,x2) = 0                                (16.3)

   q4(x1,x2,x3) = 0                                (16.4)
                ...                                (16.5)

qk2(x1,...,xn) = 0 .                               (16.6)
Here the polynomials {qj : 1 j k1} generate the same ideal that the polynomials {pj : 1 j k2} do. Therefore, the set of solutions to the collection the polynomial equations {pj = 0 : 1 j k1} equals the set of solutions to the collection of polynomial equations {qj = 0 : 1 j k2}. This canonical form greatly simplifies the task of solving the collection of polynomial equations by facilitating backsolving for xj in terms of x1,,xj-1. The effect of the ordering is to specify that variables high in the order will be eliminated while variables low in the order will not be eliminated.

In the noncommutative case, again a GB for a collection of polynomial equations is a collection of noncommuting polynomial equations in triangular form (see [HS]). There are some difficulties which don’t occur in the commutative case. For example, a GB can be infinite in the noncommutative case. However, we present software here based on the noncommutative GBA which might prove to be extremely valuable in some situations.