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XPolynomial Ring 
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The XPolynomialRing domain constructor implements generalized
polynomials with coefficients from an arbitrary Ring (not necessarily
commutative) and whose exponents are words from an arbitrary
OrderedMonoid (not necessarily commutative too).  Thus these
polynomials are (finite) linear combinations of words.

This constructor takes two arguments.  The first one is a Ring and the
second is an OrderedMonoid.  The abbreviation for XPolynomialRing is XPR.

Other constructors like XPolynomial, XRecursivePolynomial, 
XDistributedPolynomial, LiePolynomial and XPBWPolynomial implement 
multivariate polynomials in non-commutative variables.

We illustrate now some of the facilities of the XPR domain constructor.

Define the free ordered monoid generated by the symbols. ::

  Word := OrderedFreeMonoid(Symbol)
    OrderedFreeMonoid Symbol
                                 Type: Domain

Define the linear combinations of these words with integer coefficients. ::

  poly:= XPR(Integer,Word)
    XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
                                 Type: Domain

Then we define a first element from poly. ::

  p:poly := 2 * x - 3 * y + 1
    1 + 2x - 3y
                     Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)

And a second one. ::

  q:poly := 2 * x + 1
    1 + 2x
                     Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)

We compute their sum, ::

  p + q
    2 + 4x - 3y
                     Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)

their product, ::

  p * q
                    2
    1 + 4x - 3y + 4x  - 6y x
                     Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)

and see that variables do not commute. ::

  (p+q)**2-p**2-q**2-2*p*q
    - 6x y + 6y x
                     Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)

Now we define a ring of square matrices, ::

  M := SquareMatrix(2,Fraction Integer)
    SquareMatrix(2,Fraction Integer)
                                 Type: Domain

and the linear combinations of words with these  matrices as coefficients. ::

  poly1:= XPR(M,Word)
   XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
                                 Type: Domain

Define a first matrix, ::

  m1:M := matrix [ [i*j**2 for i in 1..2] for j in 1..2]
         +1  2+
         |    |
         +4  8+
                                 Type: SquareMatrix(2,Fraction Integer)

a second one, ::

  m2:M := m1 - 5/4
         +  1    +
         |- -  2 |
         |  4    |
         |       |
         |     27|
         | 4   --|
         +      4+
                                Type: SquareMatrix(2,Fraction Integer)

and a third one. ::

  m3: M := m2**2
         +129     +
         |---  13 |
         | 16     |
         |        |
         |     857|
         |26   ---|
         +      16+
                               Type: SquareMatrix(2,Fraction Integer)

Define a polynomial, ::

  pm:poly1   := m1*x + m2*y + m3*z - 2/3 
         +  2     +             +  1    +    +129     +
         |- -   0 |             |- -  2 |    |---  13 |
         |  3     |   +1  2+    |  4    |    | 16     |
         |        | + |    |x + |       |y + |        |z
         |       2|   +4  8+    |     27|    |     857|
         | 0   - -|             | 4   --|    |26   ---|
         +       3+             +      4+    +      16+
     Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),
                           OrderedFreeMonoid Symbol)

a second one, ::

  qm:poly1 := pm - m1*x
         +  2     +   +  1    +    +129     +
         |- -   0 |   |- -  2 |    |---  13 |
         |  3     |   |  4    |    | 16     |
         |        | + |       |y + |        |z
         |       2|   |     27|    |     857|
         | 0   - -|   | 4   --|    |26   ---|
         +       3+   +      4+    +      16+
      Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),
                            OrderedFreeMonoid Symbol)

and the following power. ::

  qm**3
     +   8      +   +  1  8+    +43   52 +    +  129       +
     |- --   0  |   |- -  -|    |--   -- |    |- ---  - 26 |
     |  27      |   |  3  3|    | 4    3 |    |   8        | 2
     |          | + |      |y + |        |z + |            |y
     |         8|   |16    |    |104  857|    |         857|
     | 0    - --|   |--   9|    |---  ---|    |- 52   - ---|
     +        27+   + 3    +    + 3    12+    +          8 +
   + 
     +  3199     831 +      +  3199     831 +      +  103169     6409 +
     |- ----   - --- |      |- ----   - --- |      |- ------   - ---- |
     |   32       4  |      |   32       4  |      |    128        4  | 2
     |               |y z + |               |z y + |                  |z
     |  831     26467|      |  831     26467|      |   6409     820977|
     |- ---   - -----|      |- ---   - -----|      | - ----   - ------|
     +   2        32 +      +   2        32 +      +     2        128 +
   + 
     +3199   831 +     +103169   6409 +      +103169   6409 +
     |----   --- |     |------   ---- |      |------   ---- |
     | 64     8  | 3   |  256      8  | 2    |  256      8  |
     |           |y  + |              |y z + |              |y z y
     |831   26467|     | 6409   820977|      | 6409   820977|
     |---   -----|     | ----   ------|      | ----   ------|
     + 4      64 +     +   4      256 +      +   4      256 +
   + 
     +3178239   795341 +       +103169   6409 +       +3178239   795341 +
     |-------   ------ |       |------   ---- |       |-------   ------ |
     |  1024      128  |   2   |  256      8  |   2   |  1024      128  |
     |                 |y z  + |              |z y  + |                 |z y z
     |795341   25447787|       | 6409   820977|       |795341   25447787|
     |------   --------|       | ----   ------|       |------   --------|
     +  64       1024  +       +   4      256 +       +  64       1024  +
   + 
     +3178239   795341 +      +98625409  12326223 +
     |-------   ------ |      |--------  -------- |
     |  1024      128  | 2    |  4096       256   | 3
     |                 |z y + |                   |z
     |795341   25447787|      |12326223  788893897|
     |------   --------|      |--------  ---------|
     +  64       1024  +      +   128       4096  +
        Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),
                              OrderedFreeMonoid Symbol)

See Also:

* )help XPBWPolynomial
* )help LiePolynomial
* )help XDistributedPolynomial
* )help XRecursivePolynomial
* )help XPolynomial
* )show XPolynomialRing