The key to computing ``parenthesized'' representations of a polynomial is the use of NCCollectOnVariables which has the NCAlgebra command NCCollect at its core. Before discussing the NCCollect command, we define a notion of homogeneous (noncommuting) polynomial.
Definition 9.1
Let p be a polynomial and V be a set of variables. p is
homogeneous in V if for every v
V, the number of occurences of v
in each term of p is independent of the term.
Of course,
x y z x
+
x x
is homogeneous in
{x,x},
x y z x
+
x z y
is homogeneous in
{y, z}
and but
x y z x
+
x z
is not homogeneous in
{x, x}
(since the number of
x
is the first term is 1 and
the number of
x
is the second term is 0).
When given a polynomial p and a set of variables V, NCCollect first writes the polynomial p as a sum of polynomials which are homogeneous in V. For each summand, NCCollect writes (to the extent possible) the polynomial into a ``parenthesized'' form using the rules
c
p
v +
c
p
v
p
| = |
p
v
(c
+
c
p)
|
c
p
v
p
+
c
p
v
| = |
p
v
(c
p
+
c)
|
|
c
p
v
p
+
c
v
p
| = |
(c
p
+
c)
v
p
|
c
v
p
+
c
p
v
p
| = |
(c
+
c
p)
v
p
|
|
c
p
v
p
+
c
p
v
p
| = |
p
v
(c
p
+
c
p)
|
c
p
v
p
+
c
p
v
p
| = |
(c
p
+
c
p)
v
p
|
|
c
v
+
c
v
p
| = |
v
(c
+
c
p)
|
c
v
+
c
p
v
| = |
(c
+
c
p)
v
|
where v is a variable in V,
c
and
c
are scalars,
and
p ,
p
and
p
are polynomials.
If none of the above rules apply to a V-homogeneous polynomial, then we say that its only collected form is trivial. If a polynomial is a sum of polynomials whose only collected form is trivial, then we say that this sums only collected form is trivial.