Also, even if Decompose were implementable, one would not be able to use it effectively, since the operation associates a composition of polynomials (and therefore, motivated unknowns) to a particular polynomial and applying Decompose to each polynomial in the ideal would not be practical.
For example, suppose that a polynomial
appears on a spreadsheet and has
the property that there are other polynomials
and
for which L p R
has a 1-decomposition
where k is a polynomial in one unknown.
By the definition of an ideal,
L p R
is in the ideal represented by the output of the
spreadsheet. The polynomial L p R will
not appear on the spreadsheet, since p appears on the
spreadsheet.
Therefore, in practice, a person must recognize
the L and R which yield the 1-decomposition.
We formalize this as follows.
Definition 5.1
A polynomial p motivates an unknown y via the equations y =
q(a
, ...,
a
,
x, ...,
x
)
if there exists polynomials
L(a, ...,
a,
x, ...,
x)
and
R(a, ...,
a,
x, ...,
x)
and there exists a polynomial in one unknown
k(a, ...,
a, y)
such that LpR =
k(a, ...,
a,
q(a, ...,
a,
x, ...,
x)).
Of course, from the perspective of finding zeros on collections of polynomials, if p has a zero, then L p R has a zero and so k has a zero. Since k is a polynomial in only one unknown variable, finding the zeros of k is bound to be easier than finding the zeros of p.
The definition we give here is not the most general useful definition which we could have given. A more general definition would involve finding a 1-decomposition using any polynomial in the ideal generated by p rather than those of the form L p R. We have found that the definition we give above is sufficient for the problems which we are considering.
While we do not know how to implement the Decompose operation (§), there is a certain type of ``collect'' command which we have found very helpful. This ``collect'' command assists the user in performing decompositions of the polynomial at hand and helps in finding other polynomials in the ideal which would produce motivated unknowns. (See the discussion around equations in §.)
This section gives a definition of a strategy and then describes a command which ``collects'' knowns and products of knowns out of expressions. For example, suppose that A and B are knowns and X, Y and Z are unknowns. The collected form of
(5.2) X A B X + X A B Y + Y A B X + Y A B Y + A X + A Y
is
(5.3) (X + Y) A B (X + Y) + A (X + Y).
Clearly this suggests a decomposition of (5.2) and indeed the collect command helps find decompositions of much more complicated polynomials.
Next we give a demonstration of how collect enters the NCProcess commands.