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