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.