• ### Some computations. An important thing: everything is mathematically typed in Axiom.

```(1) -> 1+1

(1)  2
Type: PositiveInteger
(2) -> integrate(1/x^(1/3),x)

3+-+2
3\|x
(2)  ------
2
Type: Union(Expression Integer,...)
```

• ### Some more complicated computations:

```)cl all

All user variables and function definitions have been cleared.

Word := OrderedFreeMonoid(Symbol)

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

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

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

(4)  1 + 2x
Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
p + q

(5)  2 + 4x - 3y
Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
p * q

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

(7)  - 6x y + 6y x
Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
M := SquareMatrix(2,Fraction Integer)

(8)  SquareMatrix(2,Fraction Integer)
Type: Domain
poly1:= XPR(M,Word)

(9)
XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
Type: Domain
m1:M := matrix [[i*j**2 for i in 1..2] for j in 1..2]

+1  2+
(10)  |    |
+4  8+
Type: SquareMatrix(2,Fraction Integer)
m2:M := m1 - 5/4

+  1    +
|- -  2 |
|  4    |
(11)  |       |
|     27|
| 4   --|
+      4+
Type: SquareMatrix(2,Fraction Integer)
m3: M := m2**2

+129     +
|---  13 |
| 16     |
(12)  |        |
|     857|
|26   ---|
+      16+
Type: SquareMatrix(2,Fraction Integer)
pm:poly1   := m1*x + m2*y + m3*z - 2/3

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

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

(15)
+   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)
```