1\begin{patch}{XPolynomialRingXmpPagePatch1}
2\begin{paste}{XPolynomialRingXmpPageFull1}{XPolynomialRingXmpPageEmpty1}
3\pastebutton{XPolynomialRingXmpPageFull1}{\hidepaste}
4\tab{5}\spadcommand{Word := FreeMonoid(Symbol)\bound{Word }}
5\indentrel{3}\begin{verbatim}
6   (1)  FreeMonoid(Symbol)
7                                             Type: Type
8\end{verbatim}
9\indentrel{-3}\end{paste}\end{patch}
10
11\begin{patch}{XPolynomialRingXmpPageEmpty1}
12\begin{paste}{XPolynomialRingXmpPageEmpty1}{XPolynomialRingXmpPagePatch1}
13\pastebutton{XPolynomialRingXmpPageEmpty1}{\showpaste}
14\tab{5}\spadcommand{Word := FreeMonoid(Symbol)\bound{Word }}
15\end{paste}\end{patch}
16
17\begin{patch}{XPolynomialRingXmpPagePatch2}
18\begin{paste}{XPolynomialRingXmpPageFull2}{XPolynomialRingXmpPageEmpty2}
19\pastebutton{XPolynomialRingXmpPageFull2}{\hidepaste}
20\tab{5}\spadcommand{poly:= XPR(Integer,Word)\free{Word }\bound{poly }}
21\indentrel{3}\begin{verbatim}
22   (2)  XPolynomialRing(Integer,FreeMonoid(Symbol))
23                                             Type: Type
24\end{verbatim}
25\indentrel{-3}\end{paste}\end{patch}
26
27\begin{patch}{XPolynomialRingXmpPageEmpty2}
28\begin{paste}{XPolynomialRingXmpPageEmpty2}{XPolynomialRingXmpPagePatch2}
29\pastebutton{XPolynomialRingXmpPageEmpty2}{\showpaste}
30\tab{5}\spadcommand{poly:= XPR(Integer,Word)\free{Word }\bound{poly }}
31\end{paste}\end{patch}
32
33\begin{patch}{XPolynomialRingXmpPagePatch3}
34\begin{paste}{XPolynomialRingXmpPageFull3}{XPolynomialRingXmpPageEmpty3}
35\pastebutton{XPolynomialRingXmpPageFull3}{\hidepaste}
36\tab{5}\spadcommand{p:poly := 2 * x - 3 * y + 1\free{poly }\bound{p }}
37\indentrel{3}\begin{verbatim}
38   (3)  1 + 2 x - 3 y
39      Type: XPolynomialRing(Integer,FreeMonoid(Symbol))
40\end{verbatim}
41\indentrel{-3}\end{paste}\end{patch}
42
43\begin{patch}{XPolynomialRingXmpPageEmpty3}
44\begin{paste}{XPolynomialRingXmpPageEmpty3}{XPolynomialRingXmpPagePatch3}
45\pastebutton{XPolynomialRingXmpPageEmpty3}{\showpaste}
46\tab{5}\spadcommand{p:poly := 2 * x - 3 * y + 1\free{poly }\bound{p }}
47\end{paste}\end{patch}
48
49\begin{patch}{XPolynomialRingXmpPagePatch4}
50\begin{paste}{XPolynomialRingXmpPageFull4}{XPolynomialRingXmpPageEmpty4}
51\pastebutton{XPolynomialRingXmpPageFull4}{\hidepaste}
52\tab{5}\spadcommand{q:poly := 2 * x + 1\free{poly }\bound{q }}
53\indentrel{3}\begin{verbatim}
54   (4)  1 + 2 x
55      Type: XPolynomialRing(Integer,FreeMonoid(Symbol))
56\end{verbatim}
57\indentrel{-3}\end{paste}\end{patch}
58
59\begin{patch}{XPolynomialRingXmpPageEmpty4}
60\begin{paste}{XPolynomialRingXmpPageEmpty4}{XPolynomialRingXmpPagePatch4}
61\pastebutton{XPolynomialRingXmpPageEmpty4}{\showpaste}
62\tab{5}\spadcommand{q:poly := 2 * x + 1\free{poly }\bound{q }}
63\end{paste}\end{patch}
64
65\begin{patch}{XPolynomialRingXmpPagePatch5}
66\begin{paste}{XPolynomialRingXmpPageFull5}{XPolynomialRingXmpPageEmpty5}
67\pastebutton{XPolynomialRingXmpPageFull5}{\hidepaste}
68\tab{5}\spadcommand{p + q\free{p }\free{q }}
69\indentrel{3}\begin{verbatim}
70   (5)  2 + 4 x - 3 y
71      Type: XPolynomialRing(Integer,FreeMonoid(Symbol))
72\end{verbatim}
73\indentrel{-3}\end{paste}\end{patch}
74
75\begin{patch}{XPolynomialRingXmpPageEmpty5}
76\begin{paste}{XPolynomialRingXmpPageEmpty5}{XPolynomialRingXmpPagePatch5}
77\pastebutton{XPolynomialRingXmpPageEmpty5}{\showpaste}
78\tab{5}\spadcommand{p + q\free{p }\free{q }}
79\end{paste}\end{patch}
80
81\begin{patch}{XPolynomialRingXmpPagePatch6}
82\begin{paste}{XPolynomialRingXmpPageFull6}{XPolynomialRingXmpPageEmpty6}
83\pastebutton{XPolynomialRingXmpPageFull6}{\hidepaste}
84\tab{5}\spadcommand{p * q\free{p }\free{q }}
85\indentrel{3}\begin{verbatim}
86                           2
87   (6)  1 + 4 x - 3 y + 4 x  - 6 y x
88      Type: XPolynomialRing(Integer,FreeMonoid(Symbol))
89\end{verbatim}
90\indentrel{-3}\end{paste}\end{patch}
91
92\begin{patch}{XPolynomialRingXmpPageEmpty6}
93\begin{paste}{XPolynomialRingXmpPageEmpty6}{XPolynomialRingXmpPagePatch6}
94\pastebutton{XPolynomialRingXmpPageEmpty6}{\showpaste}
95\tab{5}\spadcommand{p * q\free{p }\free{q }}
96\end{paste}\end{patch}
97
98\begin{patch}{XPolynomialRingXmpPagePatch7}
99\begin{paste}{XPolynomialRingXmpPageFull7}{XPolynomialRingXmpPageEmpty7}
100\pastebutton{XPolynomialRingXmpPageFull7}{\hidepaste}
101\tab{5}\spadcommand{(p +q)^2 -p^2 -q^2 - 2*p*q\free{p }\free{q }}
102\indentrel{3}\begin{verbatim}
103   (7)  - 6 x y + 6 y x
104      Type: XPolynomialRing(Integer,FreeMonoid(Symbol))
105\end{verbatim}
106\indentrel{-3}\end{paste}\end{patch}
107
108\begin{patch}{XPolynomialRingXmpPageEmpty7}
109\begin{paste}{XPolynomialRingXmpPageEmpty7}{XPolynomialRingXmpPagePatch7}
110\pastebutton{XPolynomialRingXmpPageEmpty7}{\showpaste}
111\tab{5}\spadcommand{(p +q)^2 -p^2 -q^2 - 2*p*q\free{p }\free{q }}
112\end{paste}\end{patch}
113
114\begin{patch}{XPolynomialRingXmpPagePatch8}
115\begin{paste}{XPolynomialRingXmpPageFull8}{XPolynomialRingXmpPageEmpty8}
116\pastebutton{XPolynomialRingXmpPageFull8}{\hidepaste}
117\tab{5}\spadcommand{M := SquareMatrix(2,Fraction Integer)\bound{M }}
118\indentrel{3}\begin{verbatim}
119   (8)  SquareMatrix(2,Fraction(Integer))
120                                             Type: Type
121\end{verbatim}
122\indentrel{-3}\end{paste}\end{patch}
123
124\begin{patch}{XPolynomialRingXmpPageEmpty8}
125\begin{paste}{XPolynomialRingXmpPageEmpty8}{XPolynomialRingXmpPagePatch8}
126\pastebutton{XPolynomialRingXmpPageEmpty8}{\showpaste}
127\tab{5}\spadcommand{M := SquareMatrix(2,Fraction Integer)\bound{M }}
128\end{paste}\end{patch}
129
130\begin{patch}{XPolynomialRingXmpPagePatch9}
131\begin{paste}{XPolynomialRingXmpPageFull9}{XPolynomialRingXmpPageEmpty9}
132\pastebutton{XPolynomialRingXmpPageFull9}{\hidepaste}
133\tab{5}\spadcommand{poly1:= XPR(M,Word)\free{Word }\free{M }\bound{poly1 }}
134\indentrel{3}\begin{verbatim}
135   (9)
136  XPolynomialRing(SquareMatrix(2,Fraction(Integer)),FreeM
137  onoid(Symbol))
138                                             Type: Type
139\end{verbatim}
140\indentrel{-3}\end{paste}\end{patch}
141
142\begin{patch}{XPolynomialRingXmpPageEmpty9}
143\begin{paste}{XPolynomialRingXmpPageEmpty9}{XPolynomialRingXmpPagePatch9}
144\pastebutton{XPolynomialRingXmpPageEmpty9}{\showpaste}
145\tab{5}\spadcommand{poly1:= XPR(M,Word)\free{Word }\free{M }\bound{poly1 }}
146\end{paste}\end{patch}
147
148\begin{patch}{XPolynomialRingXmpPagePatch10}
149\begin{paste}{XPolynomialRingXmpPageFull10}{XPolynomialRingXmpPageEmpty10}
150\pastebutton{XPolynomialRingXmpPageFull10}{\hidepaste}
151\tab{5}\spadcommand{m1:M := matrix [[i*j^2 for i in 1..2] for j in 1..2]\free{M }\bound{m1 }}
152\indentrel{3}\begin{verbatim}
153         +1  2+
154   (10)  |    |
155         +4  8+
156                Type: SquareMatrix(2,Fraction(Integer))
157\end{verbatim}
158\indentrel{-3}\end{paste}\end{patch}
159
160\begin{patch}{XPolynomialRingXmpPageEmpty10}
161\begin{paste}{XPolynomialRingXmpPageEmpty10}{XPolynomialRingXmpPagePatch10}
162\pastebutton{XPolynomialRingXmpPageEmpty10}{\showpaste}
163\tab{5}\spadcommand{m1:M := matrix [[i*j^2 for i in 1..2] for j in 1..2]\free{M }\bound{m1 }}
164\end{paste}\end{patch}
165
166\begin{patch}{XPolynomialRingXmpPagePatch11}
167\begin{paste}{XPolynomialRingXmpPageFull11}{XPolynomialRingXmpPageEmpty11}
168\pastebutton{XPolynomialRingXmpPageFull11}{\hidepaste}
169\tab{5}\spadcommand{m2:M := m1 - 5/4\free{M }\free{m1 }\bound{m2 }}
170\indentrel{3}\begin{verbatim}
171         +  1    +
172         |- -  2 |
173         |  4    |
174   (11)  |       |
175         |     27|
176         | 4   --|
177         +      4+
178                Type: SquareMatrix(2,Fraction(Integer))
179\end{verbatim}
180\indentrel{-3}\end{paste}\end{patch}
181
182\begin{patch}{XPolynomialRingXmpPageEmpty11}
183\begin{paste}{XPolynomialRingXmpPageEmpty11}{XPolynomialRingXmpPagePatch11}
184\pastebutton{XPolynomialRingXmpPageEmpty11}{\showpaste}
185\tab{5}\spadcommand{m2:M := m1 - 5/4\free{M }\free{m1 }\bound{m2 }}
186\end{paste}\end{patch}
187
188\begin{patch}{XPolynomialRingXmpPagePatch12}
189\begin{paste}{XPolynomialRingXmpPageFull12}{XPolynomialRingXmpPageEmpty12}
190\pastebutton{XPolynomialRingXmpPageFull12}{\hidepaste}
191\tab{5}\spadcommand{m3: M := m2^2\free{M }\free{m2 }\bound{m3 }}
192\indentrel{3}\begin{verbatim}
193         +129     +
194         |---  13 |
195         | 16     |
196   (12)  |        |
197         |     857|
198         |26   ---|
199         +      16+
200                Type: SquareMatrix(2,Fraction(Integer))
201\end{verbatim}
202\indentrel{-3}\end{paste}\end{patch}
203
204\begin{patch}{XPolynomialRingXmpPageEmpty12}
205\begin{paste}{XPolynomialRingXmpPageEmpty12}{XPolynomialRingXmpPagePatch12}
206\pastebutton{XPolynomialRingXmpPageEmpty12}{\showpaste}
207\tab{5}\spadcommand{m3: M := m2^2\free{M }\free{m2 }\bound{m3 }}
208\end{paste}\end{patch}
209
210\begin{patch}{XPolynomialRingXmpPagePatch13}
211\begin{paste}{XPolynomialRingXmpPageFull13}{XPolynomialRingXmpPageEmpty13}
212\pastebutton{XPolynomialRingXmpPageFull13}{\hidepaste}
213\tab{5}\spadcommand{pm:poly1 := m1*x + m2*y + m3*z - 2/3\free{poly1 }\free{m1 }\free{m2 }\free{m3 }\bound{pm }}
214\indentrel{3}\begin{verbatim}
215         +  2     +             +  1    +    +129     +
216         |- -   0 |             |- -  2 |    |---  13 |
217         |  3     |   +1  2+    |  4    |    | 16     |
218   (13)  |        | + |    |x + |       |y + |        |z
219         |       2|   +4  8+    |     27|    |     857|
220         | 0   - -|             | 4   --|    |26   ---|
221         +       3+             +      4+    +      16+
222Type: XPolynomialRing(SquareMatrix(2,Fraction(Integer)),FreeMonoid(Symbol))
223\end{verbatim}
224\indentrel{-3}\end{paste}\end{patch}
225
226\begin{patch}{XPolynomialRingXmpPageEmpty13}
227\begin{paste}{XPolynomialRingXmpPageEmpty13}{XPolynomialRingXmpPagePatch13}
228\pastebutton{XPolynomialRingXmpPageEmpty13}{\showpaste}
229\tab{5}\spadcommand{pm:poly1 := m1*x + m2*y + m3*z - 2/3\free{poly1 }\free{m1 }\free{m2 }\free{m3 }\bound{pm }}
230\end{paste}\end{patch}
231
232\begin{patch}{XPolynomialRingXmpPagePatch14}
233\begin{paste}{XPolynomialRingXmpPageFull14}{XPolynomialRingXmpPageEmpty14}
234\pastebutton{XPolynomialRingXmpPageFull14}{\hidepaste}
235\tab{5}\spadcommand{qm:poly1 := pm - m1*x\free{m1 }\free{pm }\bound{qm }}
236\indentrel{3}\begin{verbatim}
237         +  2     +   +  1    +    +129     +
238         |- -   0 |   |- -  2 |    |---  13 |
239         |  3     |   |  4    |    | 16     |
240   (14)  |        | + |       |y + |        |z
241         |       2|   |     27|    |     857|
242         | 0   - -|   | 4   --|    |26   ---|
243         +       3+   +      4+    +      16+
244Type: XPolynomialRing(SquareMatrix(2,Fraction(Integer)),FreeMonoid(Symbol))
245\end{verbatim}
246\indentrel{-3}\end{paste}\end{patch}
247
248\begin{patch}{XPolynomialRingXmpPageEmpty14}
249\begin{paste}{XPolynomialRingXmpPageEmpty14}{XPolynomialRingXmpPagePatch14}
250\pastebutton{XPolynomialRingXmpPageEmpty14}{\showpaste}
251\tab{5}\spadcommand{qm:poly1 := pm - m1*x\free{m1 }\free{pm }\bound{qm }}
252\end{paste}\end{patch}
253
254\begin{patch}{XPolynomialRingXmpPagePatch15}
255\begin{paste}{XPolynomialRingXmpPageFull15}{XPolynomialRingXmpPageEmpty15}
256\pastebutton{XPolynomialRingXmpPageFull15}{\hidepaste}
257\tab{5}\spadcommand{qm^3\bound{qm }}
258\indentrel{3}\begin{verbatim}
259   (15)
260     +   8      +   +  1  8+    +43   52 +
261     |- --   0  |   |- -  -|    |--   -- |
262     |  27      |   |  3  3|    | 4    3 |
263     |          | + |      |y + |        |z
264     |         8|   |16    |    |104  857|
265     | 0    - --|   |--   9|    |---  ---|
266     +        27+   + 3    +    + 3    12+
267   +
268     +  129       +     +  3199     831 +
269     |- ---  - 26 |     |- ----   - --- |
270     |   8        | 2   |   32       4  |
271     |            |y  + |               |y z
272     |         857|     |  831     26467|
273     |- 52   - ---|     |- ---   - -----|
274     +          8 +     +   2        32 +
275   +
276     +  3199     831 +      +  103169     6409 +
277     |- ----   - --- |      |- ------   - ---- |
278     |   32       4  |      |    128        4  | 2
279     |               |z y + |                  |z
280     |  831     26467|      |   6409     820977|
281     |- ---   - -----|      | - ----   - ------|
282     +   2        32 +      +     2        128 +
283   +
284     +3199   831 +     +103169   6409 +
285     |----   --- |     |------   ---- |
286     | 64     8  | 3   |  256      8  | 2
287     |           |y  + |              |y z
288     |831   26467|     | 6409   820977|
289     |---   -----|     | ----   ------|
290     + 4      64 +     +   4      256 +
291   +
292     +103169   6409 +        +3178239   795341 +
293     |------   ---- |        |-------   ------ |
294     |  256      8  |        |  1024      128  |   2
295     |              |y z y + |                 |y z
296     | 6409   820977|        |795341   25447787|
297     | ----   ------|        |------   --------|
298     +   4      256 +        +  64       1024  +
299   +
300     +103169   6409 +       +3178239   795341 +
301     |------   ---- |       |-------   ------ |
302     |  256      8  |   2   |  1024      128  |
303     |              |z y  + |                 |z y z
304     | 6409   820977|       |795341   25447787|
305     | ----   ------|       |------   --------|
306     +   4      256 +       +  64       1024  +
307   +
308     +3178239   795341 +      +98625409  12326223 +
309     |-------   ------ |      |--------  -------- |
310     |  1024      128  | 2    |  4096       256   | 3
311     |                 |z y + |                   |z
312     |795341   25447787|      |12326223  788893897|
313     |------   --------|      |--------  ---------|
314     +  64       1024  +      +   128       4096  +
315Type: XPolynomialRing(SquareMatrix(2,Fraction(Integer)),FreeMonoid(Symbol))
316\end{verbatim}
317\indentrel{-3}\end{paste}\end{patch}
318
319\begin{patch}{XPolynomialRingXmpPageEmpty15}
320\begin{paste}{XPolynomialRingXmpPageEmpty15}{XPolynomialRingXmpPagePatch15}
321\pastebutton{XPolynomialRingXmpPageEmpty15}{\showpaste}
322\tab{5}\spadcommand{qm^3\bound{qm }}
323\end{paste}\end{patch}
324
325