This is valuable since the user will ordinarily know that a polynomial of a very high degree will not be useful to him and so there is no reason to produce it. It is not the time that it takes to produce a large polynomial that is the primary factor. Rather it is the reduction algorithms that will get bogged down trying to remove it. Degree caps prevent the algorithm from ever producing polynomials over a certain degree, or combining polynomials over a certain degree, and the user will still be left with a generating set for the ideal generated by the input equations. There are two different options associated with degree caps. For instance,

DegreeCap 8

would prevent a polynomial of degree 8 or higher from combining with a polynomial of higher degree.

DegreeSumCap 10

would prevent two polynomials whose degrees add up to 10 or more from combining. Degree caps could prevent an important relation from being created, so when there is a lack of progress, raising the degree caps as well as the iteration number would be the next step.

**WE URGE USE OF DEGREE CAPS. THEY SAVE A LOT OF TIME.**

Wed Jul 3 10:27:42 PDT 1996