Preview of operations

Example 12.1 shows that, in some cases, we seek a minimal generating set for an ideal; Example 12.3 shows that in some cases we prefer certain minimal generating sets for an ideal to other minimal generating sets; Example 12.4 shows that in some cases we prefer finding small generating sets. The choice of how to shrink a generating set is further complicated by the desire to have shrinking occur quickly on the computer.

The remainder of this section describes six different operations to convert from a particular basis for an ideal to a smaller one. They differ in approach, speed and functionality. We now give a brief description of each of these six operations.

(1) SmallBasis can be used to find smaller generating sets and is relatively fast. SmallBasis is described in §. SmallBasis uses the GBA heavily.

(2) ShrinkBasis is used to find all minimal generating sets and can be very slow. ShrinkBasis is often too slow to use in practice. ShrinkBasis is described in §. ShrinkBasis uses the GBA heavily.

(3) RemoveRedundant is used to find smaller generating sets and is very fast. The RemoveRedundant operation is described in §. The Remove Redundant operation requires the recording of information during the previous run of the GBA and uses that information once the run is completed. For this reason, the explanation of Remove Redundant requires some initial theoretical discussions which are in §. The run time of Remove Redundant is very fast since it implements a graph search and does not invoke a GBA.

(4,5) The fourth and fifth operations are SmallBasis By Category and Remove Redundant By Category. These two operations find smaller subsets while respecting the fact that retaining polynomials which do not involve unknowns is desirable and they act on each category individually. These last two operations are described in §. These two operations use SmallBasis and RemoveRedundant, respectively, heavily.

(6) The sixth operation is RemoveRedundantProtect. This command combines the results of three Remove Redundant runs and is very fast. This command is used to prevent equations not involving unknowns from being removed by polynomials which involve unknowns. Also, the command prevents digested polynomials from being removed by polynomials which are undigested. This command is described in §.

Recall, for the remainder of the section, that the ideal in generated by is denoted .