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Introducing a new motivated unknown

As in the introduction (§), suppose we are in a context where there are knowns tex2html_wrap_inline4194 and unknowns tex2html_wrap_inline4196 . 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 tex2html_wrap_inline5520 tex2html_wrap_inline5522 . 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 tex2html_wrap_inline5528 and tex2html_wrap_inline5530 , set tex2html_wrap_inline5532 and tex2html_wrap_inline5534 lower in the ordering than all other unknowns and run the GBA. This will convert (5.7) to


If tex2html_wrap_inline5538 contains no unknowns, then we have found the desired decomposition.

For example, if, in a computer session, A, B, tex2html_wrap_inline5482 and tex2html_wrap_inline5546 are set to be knowns and b, d, tex2html_wrap_inline5552 and tex2html_wrap_inline5292 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 tex2html_wrap_inline5556 and tex2html_wrap_inline5558 . Step 3 above converts (5.8) to


which gives the decomposition tex2html_wrap_inline5560 for p as described in Example 5.4.

Wed Jul 3 10:27:42 PDT 1996