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## Computing decompositions

In this section we give examples of how one implements the 1-Decompose operation with human intervention.

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.

Helton
Wed Jul 3 10:27:42 PDT 1996