We use an example to remind the reader of the notion of a decomposition of a polynomial (see §).
Example 5.4 Suppose A, A, B and B are knowns and b, d, b and d are unknown. Consider the polynomial
|(5.5)||p(A, A, B, B, b, b, d, d) = 1 - B B - d d + B b d - B b b B + d b B + d A A d - B b A A d - d A A b B + B b A A b B|
It can be written as
|(5.6)||p = 1 - B B - (d - B b)( d - b B) + (d - B b) A A (d - b B)|
where q = d - B b. Note that k is a polynomial whose only unknowns are q and q.
We do not know a way to compute decompositions and hopefully this paper's need for a decomposition operator will stimulate a serious quest for a way to compute decompositions. For now, the best we can hope for is a modest amount of computer assistance which is why we turn to the collect-type commands of the next section.