Introducing a new motivated unknown

As in the introduction (§), suppose we are in a context where there are knowns and unknowns . We now describe how we shall be using the NCCV command to help discover decompositions. (In the forthcoming example (see §), Step 3 below is easy enough that it is done by inspection. No run of the GBA is required.) NCCV can be used as follows:

(1) Apply NCCV to a polynomial p and pay particular attention to the terms containing the most knowns. That is, for each term, compute the degree of that term with respect to the set of knowns and use the ones with the highest degree in this sense.

(2) The collected form of p may suggest a decomposition of p or suggest that there is another polynomial in the ideal which has a decomposition. Let us suppose that one of the the parenthesized summands of the collected form of p has a decomposition as . Therefore, we obtain

(5.7)           p = k(a, ..., a, q (a, ..., a, x, ..., x), q (a, ..., a, x, ..., x)) + s

from this operation where k is a polynomial and s is a polynomial.

(3) Declare both of the new variables and , set and lower in the ordering than all other unknowns and run the GBA. This will convert (5.7) to

If contains no unknowns, then we have found the desired decomposition.

For example, if, in a computer session, A, B, and are set to be knowns and b, d, and are set to be unknowns, then

(5.8)           1 - d d + B (b d - (1 + b b) B) + d b B - (d - B b) A A (b B - d)

is the output of NCCollectOnVariables when applied to (5.5). This suggests we set and . Step 3 above converts (5.8) to

which gives the decomposition for p as described in Example 5.4.