1\begin{patch}{PolynomialXmpPagePatch1} 2\begin{paste}{PolynomialXmpPageFull1}{PolynomialXmpPageEmpty1} 3\pastebutton{PolynomialXmpPageFull1}{\hidepaste} 4\tab{5}\spadcommand{x + 1} 5\indentrel{3}\begin{verbatim} 6 (1) x + 1 7 Type: Polynomial(Integer) 8\end{verbatim} 9\indentrel{-3}\end{paste}\end{patch} 10 11\begin{patch}{PolynomialXmpPageEmpty1} 12\begin{paste}{PolynomialXmpPageEmpty1}{PolynomialXmpPagePatch1} 13\pastebutton{PolynomialXmpPageEmpty1}{\showpaste} 14\tab{5}\spadcommand{x + 1} 15\end{paste}\end{patch} 16 17\begin{patch}{PolynomialXmpPagePatch2} 18\begin{paste}{PolynomialXmpPageFull2}{PolynomialXmpPageEmpty2} 19\pastebutton{PolynomialXmpPageFull2}{\hidepaste} 20\tab{5}\spadcommand{z - 2.3} 21\indentrel{3}\begin{verbatim} 22 (2) z - 2.3 23 Type: Polynomial(Float) 24\end{verbatim} 25\indentrel{-3}\end{paste}\end{patch} 26 27\begin{patch}{PolynomialXmpPageEmpty2} 28\begin{paste}{PolynomialXmpPageEmpty2}{PolynomialXmpPagePatch2} 29\pastebutton{PolynomialXmpPageEmpty2}{\showpaste} 30\tab{5}\spadcommand{z - 2.3} 31\end{paste}\end{patch} 32 33\begin{patch}{PolynomialXmpPagePatch3} 34\begin{paste}{PolynomialXmpPageFull3}{PolynomialXmpPageEmpty3} 35\pastebutton{PolynomialXmpPageFull3}{\hidepaste} 36\tab{5}\spadcommand{y^2 - z + 3/4} 37\indentrel{3}\begin{verbatim} 38 2 3 39 (3) - z + y + - 40 4 41 Type: Polynomial(Fraction(Integer)) 42\end{verbatim} 43\indentrel{-3}\end{paste}\end{patch} 44 45\begin{patch}{PolynomialXmpPageEmpty3} 46\begin{paste}{PolynomialXmpPageEmpty3}{PolynomialXmpPagePatch3} 47\pastebutton{PolynomialXmpPageEmpty3}{\showpaste} 48\tab{5}\spadcommand{y^2 - z + 3/4} 49\end{paste}\end{patch} 50 51\begin{patch}{PolynomialXmpPagePatch4} 52\begin{paste}{PolynomialXmpPageFull4}{PolynomialXmpPageEmpty4} 53\pastebutton{PolynomialXmpPageFull4}{\hidepaste} 54\tab{5}\spadcommand{y ^2 + x*y + y\bound{prev }} 55\indentrel{3}\begin{verbatim} 56 2 57 (4) y + (x + 1)y 58 Type: Polynomial(Integer) 59\end{verbatim} 60\indentrel{-3}\end{paste}\end{patch} 61 62\begin{patch}{PolynomialXmpPageEmpty4} 63\begin{paste}{PolynomialXmpPageEmpty4}{PolynomialXmpPagePatch4} 64\pastebutton{PolynomialXmpPageEmpty4}{\showpaste} 65\tab{5}\spadcommand{y ^2 + x*y + y\bound{prev }} 66\end{paste}\end{patch} 67 68\begin{patch}{PolynomialXmpPagePatch5} 69\begin{paste}{PolynomialXmpPageFull5}{PolynomialXmpPageEmpty5} 70\pastebutton{PolynomialXmpPageFull5}{\hidepaste} 71\tab{5}\spadcommand{\% :: DMP([y,x],INT)\free{prev }} 72\indentrel{3}\begin{verbatim} 73 2 74 (5) y + y x + y 75 Type: DistributedMultivariatePolynomial([y,x],Integer) 76\end{verbatim} 77\indentrel{-3}\end{paste}\end{patch} 78 79\begin{patch}{PolynomialXmpPageEmpty5} 80\begin{paste}{PolynomialXmpPageEmpty5}{PolynomialXmpPagePatch5} 81\pastebutton{PolynomialXmpPageEmpty5}{\showpaste} 82\tab{5}\spadcommand{\% :: DMP([y,x],INT)\free{prev }} 83\end{paste}\end{patch} 84 85\begin{patch}{PolynomialXmpPagePatch6} 86\begin{paste}{PolynomialXmpPageFull6}{PolynomialXmpPageEmpty6} 87\pastebutton{PolynomialXmpPageFull6}{\hidepaste} 88\tab{5}\spadcommand{p := (y-1)^2 * x * z\bound{p }} 89\indentrel{3}\begin{verbatim} 90 2 91 (6) (x y - 2 x y + x)z 92 Type: Polynomial(Integer) 93\end{verbatim} 94\indentrel{-3}\end{paste}\end{patch} 95 96\begin{patch}{PolynomialXmpPageEmpty6} 97\begin{paste}{PolynomialXmpPageEmpty6}{PolynomialXmpPagePatch6} 98\pastebutton{PolynomialXmpPageEmpty6}{\showpaste} 99\tab{5}\spadcommand{p := (y-1)^2 * x * z\bound{p }} 100\end{paste}\end{patch} 101 102\begin{patch}{PolynomialXmpPagePatch7} 103\begin{paste}{PolynomialXmpPageFull7}{PolynomialXmpPageEmpty7} 104\pastebutton{PolynomialXmpPageFull7}{\hidepaste} 105\tab{5}\spadcommand{q := (y-1) * x * (z+5)\bound{q }} 106\indentrel{3}\begin{verbatim} 107 (7) (x y - x)z + 5 x y - 5 x 108 Type: Polynomial(Integer) 109\end{verbatim} 110\indentrel{-3}\end{paste}\end{patch} 111 112\begin{patch}{PolynomialXmpPageEmpty7} 113\begin{paste}{PolynomialXmpPageEmpty7}{PolynomialXmpPagePatch7} 114\pastebutton{PolynomialXmpPageEmpty7}{\showpaste} 115\tab{5}\spadcommand{q := (y-1) * x * (z+5)\bound{q }} 116\end{paste}\end{patch} 117 118\begin{patch}{PolynomialXmpPagePatch8} 119\begin{paste}{PolynomialXmpPageFull8}{PolynomialXmpPageEmpty8} 120\pastebutton{PolynomialXmpPageFull8}{\hidepaste} 121\tab{5}\spadcommand{factor(q)\free{q }} 122\indentrel{3}\begin{verbatim} 123 (8) x(y - 1)(z + 5) 124 Type: Factored(Polynomial(Integer)) 125\end{verbatim} 126\indentrel{-3}\end{paste}\end{patch} 127 128\begin{patch}{PolynomialXmpPageEmpty8} 129\begin{paste}{PolynomialXmpPageEmpty8}{PolynomialXmpPagePatch8} 130\pastebutton{PolynomialXmpPageEmpty8}{\showpaste} 131\tab{5}\spadcommand{factor(q)\free{q }} 132\end{paste}\end{patch} 133 134\begin{patch}{PolynomialXmpPagePatch9} 135\begin{paste}{PolynomialXmpPageFull9}{PolynomialXmpPageEmpty9} 136\pastebutton{PolynomialXmpPageFull9}{\hidepaste} 137\tab{5}\spadcommand{p - q^2\free{p q }} 138\indentrel{3}\begin{verbatim} 139 (9) 140 2 2 2 2 2 141 (- x y + 2 x y - x )z 142 + 143 2 2 2 2 144 ((- 10 x + x)y + (20 x - 2 x)y - 10 x + x)z 145 + 146 2 2 2 2 147 - 25 x y + 50 x y - 25 x 148 Type: Polynomial(Integer) 149\end{verbatim} 150\indentrel{-3}\end{paste}\end{patch} 151 152\begin{patch}{PolynomialXmpPageEmpty9} 153\begin{paste}{PolynomialXmpPageEmpty9}{PolynomialXmpPagePatch9} 154\pastebutton{PolynomialXmpPageEmpty9}{\showpaste} 155\tab{5}\spadcommand{p - q^2\free{p q }} 156\end{paste}\end{patch} 157 158\begin{patch}{PolynomialXmpPagePatch10} 159\begin{paste}{PolynomialXmpPageFull10}{PolynomialXmpPageEmpty10} 160\pastebutton{PolynomialXmpPageFull10}{\hidepaste} 161\tab{5}\spadcommand{gcd(p,q)\free{p q }\bound{prev4 }} 162\indentrel{3}\begin{verbatim} 163 (10) x y - x 164 Type: Polynomial(Integer) 165\end{verbatim} 166\indentrel{-3}\end{paste}\end{patch} 167 168\begin{patch}{PolynomialXmpPageEmpty10} 169\begin{paste}{PolynomialXmpPageEmpty10}{PolynomialXmpPagePatch10} 170\pastebutton{PolynomialXmpPageEmpty10}{\showpaste} 171\tab{5}\spadcommand{gcd(p,q)\free{p q }\bound{prev4 }} 172\end{paste}\end{patch} 173 174\begin{patch}{PolynomialXmpPagePatch11} 175\begin{paste}{PolynomialXmpPageFull11}{PolynomialXmpPageEmpty11} 176\pastebutton{PolynomialXmpPageFull11}{\hidepaste} 177\tab{5}\spadcommand{factor \%\free{prev4 }} 178\indentrel{3}\begin{verbatim} 179 (11) x(y - 1) 180 Type: Factored(Polynomial(Integer)) 181\end{verbatim} 182\indentrel{-3}\end{paste}\end{patch} 183 184\begin{patch}{PolynomialXmpPageEmpty11} 185\begin{paste}{PolynomialXmpPageEmpty11}{PolynomialXmpPagePatch11} 186\pastebutton{PolynomialXmpPageEmpty11}{\showpaste} 187\tab{5}\spadcommand{factor \%\free{prev4 }} 188\end{paste}\end{patch} 189 190\begin{patch}{PolynomialXmpPagePatch12} 191\begin{paste}{PolynomialXmpPageFull12}{PolynomialXmpPageEmpty12} 192\pastebutton{PolynomialXmpPageFull12}{\hidepaste} 193\tab{5}\spadcommand{lcm(p,q)\free{p q }} 194\indentrel{3}\begin{verbatim} 195 2 2 2 196 (12) (x y - 2 x y + x)z + (5 x y - 10 x y + 5 x)z 197 Type: Polynomial(Integer) 198\end{verbatim} 199\indentrel{-3}\end{paste}\end{patch} 200 201\begin{patch}{PolynomialXmpPageEmpty12} 202\begin{paste}{PolynomialXmpPageEmpty12}{PolynomialXmpPagePatch12} 203\pastebutton{PolynomialXmpPageEmpty12}{\showpaste} 204\tab{5}\spadcommand{lcm(p,q)\free{p q }} 205\end{paste}\end{patch} 206 207\begin{patch}{PolynomialXmpPagePatch13} 208\begin{paste}{PolynomialXmpPageFull13}{PolynomialXmpPageEmpty13} 209\pastebutton{PolynomialXmpPageFull13}{\hidepaste} 210\tab{5}\spadcommand{content p\free{p }} 211\indentrel{3}\begin{verbatim} 212 (13) 1 213 Type: PositiveInteger 214\end{verbatim} 215\indentrel{-3}\end{paste}\end{patch} 216 217\begin{patch}{PolynomialXmpPageEmpty13} 218\begin{paste}{PolynomialXmpPageEmpty13}{PolynomialXmpPagePatch13} 219\pastebutton{PolynomialXmpPageEmpty13}{\showpaste} 220\tab{5}\spadcommand{content p\free{p }} 221\end{paste}\end{patch} 222 223\begin{patch}{PolynomialXmpPagePatch14} 224\begin{paste}{PolynomialXmpPageFull14}{PolynomialXmpPageEmpty14} 225\pastebutton{PolynomialXmpPageFull14}{\hidepaste} 226\tab{5}\spadcommand{resultant(p,q,z)\free{p q }} 227\indentrel{3}\begin{verbatim} 228 2 3 2 2 2 2 229 (14) 5 x y - 15 x y + 15 x y - 5 x 230 Type: Polynomial(Integer) 231\end{verbatim} 232\indentrel{-3}\end{paste}\end{patch} 233 234\begin{patch}{PolynomialXmpPageEmpty14} 235\begin{paste}{PolynomialXmpPageEmpty14}{PolynomialXmpPagePatch14} 236\pastebutton{PolynomialXmpPageEmpty14}{\showpaste} 237\tab{5}\spadcommand{resultant(p,q,z)\free{p q }} 238\end{paste}\end{patch} 239 240\begin{patch}{PolynomialXmpPagePatch15} 241\begin{paste}{PolynomialXmpPageFull15}{PolynomialXmpPageEmpty15} 242\pastebutton{PolynomialXmpPageFull15}{\hidepaste} 243\tab{5}\spadcommand{resultant(p,q,x)\free{p }\free{q }} 244\indentrel{3}\begin{verbatim} 245 (15) 0 246 Type: Polynomial(Integer) 247\end{verbatim} 248\indentrel{-3}\end{paste}\end{patch} 249 250\begin{patch}{PolynomialXmpPageEmpty15} 251\begin{paste}{PolynomialXmpPageEmpty15}{PolynomialXmpPagePatch15} 252\pastebutton{PolynomialXmpPageEmpty15}{\showpaste} 253\tab{5}\spadcommand{resultant(p,q,x)\free{p }\free{q }} 254\end{paste}\end{patch} 255 256\begin{patch}{PolynomialXmpPagePatch16} 257\begin{paste}{PolynomialXmpPageFull16}{PolynomialXmpPageEmpty16} 258\pastebutton{PolynomialXmpPageFull16}{\hidepaste} 259\tab{5}\spadcommand{mainVariable p\free{p }} 260\indentrel{3}\begin{verbatim} 261 (16) z 262 Type: Union(Symbol,...) 263\end{verbatim} 264\indentrel{-3}\end{paste}\end{patch} 265 266\begin{patch}{PolynomialXmpPageEmpty16} 267\begin{paste}{PolynomialXmpPageEmpty16}{PolynomialXmpPagePatch16} 268\pastebutton{PolynomialXmpPageEmpty16}{\showpaste} 269\tab{5}\spadcommand{mainVariable p\free{p }} 270\end{paste}\end{patch} 271 272\begin{patch}{PolynomialXmpPagePatch17} 273\begin{paste}{PolynomialXmpPageFull17}{PolynomialXmpPageEmpty17} 274\pastebutton{PolynomialXmpPageFull17}{\hidepaste} 275\tab{5}\spadcommand{mainVariable(1 :: POLY INT)} 276\indentrel{3}\begin{verbatim} 277 (17) "failed" 278 Type: Union("failed",...) 279\end{verbatim} 280\indentrel{-3}\end{paste}\end{patch} 281 282\begin{patch}{PolynomialXmpPageEmpty17} 283\begin{paste}{PolynomialXmpPageEmpty17}{PolynomialXmpPagePatch17} 284\pastebutton{PolynomialXmpPageEmpty17}{\showpaste} 285\tab{5}\spadcommand{mainVariable(1 :: POLY INT)} 286\end{paste}\end{patch} 287 288\begin{patch}{PolynomialXmpPagePatch18} 289\begin{paste}{PolynomialXmpPageFull18}{PolynomialXmpPageEmpty18} 290\pastebutton{PolynomialXmpPageFull18}{\hidepaste} 291\tab{5}\spadcommand{ground? p\free{p }} 292\indentrel{3}\begin{verbatim} 293 (18) false 294 Type: Boolean 295\end{verbatim} 296\indentrel{-3}\end{paste}\end{patch} 297 298\begin{patch}{PolynomialXmpPageEmpty18} 299\begin{paste}{PolynomialXmpPageEmpty18}{PolynomialXmpPagePatch18} 300\pastebutton{PolynomialXmpPageEmpty18}{\showpaste} 301\tab{5}\spadcommand{ground? p\free{p }} 302\end{paste}\end{patch} 303 304\begin{patch}{PolynomialXmpPagePatch19} 305\begin{paste}{PolynomialXmpPageFull19}{PolynomialXmpPageEmpty19} 306\pastebutton{PolynomialXmpPageFull19}{\hidepaste} 307\tab{5}\spadcommand{ground?(1 :: POLY INT)} 308\indentrel{3}\begin{verbatim} 309 (19) true 310 Type: Boolean 311\end{verbatim} 312\indentrel{-3}\end{paste}\end{patch} 313 314\begin{patch}{PolynomialXmpPageEmpty19} 315\begin{paste}{PolynomialXmpPageEmpty19}{PolynomialXmpPagePatch19} 316\pastebutton{PolynomialXmpPageEmpty19}{\showpaste} 317\tab{5}\spadcommand{ground?(1 :: POLY INT)} 318\end{paste}\end{patch} 319 320\begin{patch}{PolynomialXmpPagePatch20} 321\begin{paste}{PolynomialXmpPageFull20}{PolynomialXmpPageEmpty20} 322\pastebutton{PolynomialXmpPageFull20}{\hidepaste} 323\tab{5}\spadcommand{variables p\free{p }} 324\indentrel{3}\begin{verbatim} 325 (20) [z, y, x] 326 Type: List(Symbol) 327\end{verbatim} 328\indentrel{-3}\end{paste}\end{patch} 329 330\begin{patch}{PolynomialXmpPageEmpty20} 331\begin{paste}{PolynomialXmpPageEmpty20}{PolynomialXmpPagePatch20} 332\pastebutton{PolynomialXmpPageEmpty20}{\showpaste} 333\tab{5}\spadcommand{variables p\free{p }} 334\end{paste}\end{patch} 335 336\begin{patch}{PolynomialXmpPagePatch21} 337\begin{paste}{PolynomialXmpPageFull21}{PolynomialXmpPageEmpty21} 338\pastebutton{PolynomialXmpPageFull21}{\hidepaste} 339\tab{5}\spadcommand{degree(p,x)\free{p }} 340\indentrel{3}\begin{verbatim} 341 (21) 1 342 Type: PositiveInteger 343\end{verbatim} 344\indentrel{-3}\end{paste}\end{patch} 345 346\begin{patch}{PolynomialXmpPageEmpty21} 347\begin{paste}{PolynomialXmpPageEmpty21}{PolynomialXmpPagePatch21} 348\pastebutton{PolynomialXmpPageEmpty21}{\showpaste} 349\tab{5}\spadcommand{degree(p,x)\free{p }} 350\end{paste}\end{patch} 351 352\begin{patch}{PolynomialXmpPagePatch22} 353\begin{paste}{PolynomialXmpPageFull22}{PolynomialXmpPageEmpty22} 354\pastebutton{PolynomialXmpPageFull22}{\hidepaste} 355\tab{5}\spadcommand{degree(p,y)\free{p }} 356\indentrel{3}\begin{verbatim} 357 (22) 2 358 Type: PositiveInteger 359\end{verbatim} 360\indentrel{-3}\end{paste}\end{patch} 361 362\begin{patch}{PolynomialXmpPageEmpty22} 363\begin{paste}{PolynomialXmpPageEmpty22}{PolynomialXmpPagePatch22} 364\pastebutton{PolynomialXmpPageEmpty22}{\showpaste} 365\tab{5}\spadcommand{degree(p,y)\free{p }} 366\end{paste}\end{patch} 367 368\begin{patch}{PolynomialXmpPagePatch23} 369\begin{paste}{PolynomialXmpPageFull23}{PolynomialXmpPageEmpty23} 370\pastebutton{PolynomialXmpPageFull23}{\hidepaste} 371\tab{5}\spadcommand{degree(p,z)\free{p }} 372\indentrel{3}\begin{verbatim} 373 (23) 1 374 Type: PositiveInteger 375\end{verbatim} 376\indentrel{-3}\end{paste}\end{patch} 377 378\begin{patch}{PolynomialXmpPageEmpty23} 379\begin{paste}{PolynomialXmpPageEmpty23}{PolynomialXmpPagePatch23} 380\pastebutton{PolynomialXmpPageEmpty23}{\showpaste} 381\tab{5}\spadcommand{degree(p,z)\free{p }} 382\end{paste}\end{patch} 383 384\begin{patch}{PolynomialXmpPagePatch24} 385\begin{paste}{PolynomialXmpPageFull24}{PolynomialXmpPageEmpty24} 386\pastebutton{PolynomialXmpPageFull24}{\hidepaste} 387\tab{5}\spadcommand{degree(p,[x,y,z])\free{p }} 388\indentrel{3}\begin{verbatim} 389 (24) [1, 2, 1] 390 Type: List(NonNegativeInteger) 391\end{verbatim} 392\indentrel{-3}\end{paste}\end{patch} 393 394\begin{patch}{PolynomialXmpPageEmpty24} 395\begin{paste}{PolynomialXmpPageEmpty24}{PolynomialXmpPagePatch24} 396\pastebutton{PolynomialXmpPageEmpty24}{\showpaste} 397\tab{5}\spadcommand{degree(p,[x,y,z])\free{p }} 398\end{paste}\end{patch} 399 400\begin{patch}{PolynomialXmpPagePatch25} 401\begin{paste}{PolynomialXmpPageFull25}{PolynomialXmpPageEmpty25} 402\pastebutton{PolynomialXmpPageFull25}{\hidepaste} 403\tab{5}\spadcommand{minimumDegree(p,z)\free{p }} 404\indentrel{3}\begin{verbatim} 405 (25) 1 406 Type: PositiveInteger 407\end{verbatim} 408\indentrel{-3}\end{paste}\end{patch} 409 410\begin{patch}{PolynomialXmpPageEmpty25} 411\begin{paste}{PolynomialXmpPageEmpty25}{PolynomialXmpPagePatch25} 412\pastebutton{PolynomialXmpPageEmpty25}{\showpaste} 413\tab{5}\spadcommand{minimumDegree(p,z)\free{p }} 414\end{paste}\end{patch} 415 416\begin{patch}{PolynomialXmpPagePatch26} 417\begin{paste}{PolynomialXmpPageFull26}{PolynomialXmpPageEmpty26} 418\pastebutton{PolynomialXmpPageFull26}{\hidepaste} 419\tab{5}\spadcommand{totalDegree p\free{p }} 420\indentrel{3}\begin{verbatim} 421 (26) 4 422 Type: PositiveInteger 423\end{verbatim} 424\indentrel{-3}\end{paste}\end{patch} 425 426\begin{patch}{PolynomialXmpPageEmpty26} 427\begin{paste}{PolynomialXmpPageEmpty26}{PolynomialXmpPagePatch26} 428\pastebutton{PolynomialXmpPageEmpty26}{\showpaste} 429\tab{5}\spadcommand{totalDegree p\free{p }} 430\end{paste}\end{patch} 431 432\begin{patch}{PolynomialXmpPagePatch27} 433\begin{paste}{PolynomialXmpPageFull27}{PolynomialXmpPageEmpty27} 434\pastebutton{PolynomialXmpPageFull27}{\hidepaste} 435\tab{5}\spadcommand{leadingMonomial p\free{p }} 436\indentrel{3}\begin{verbatim} 437 2 438 (27) x y z 439 Type: Polynomial(Integer) 440\end{verbatim} 441\indentrel{-3}\end{paste}\end{patch} 442 443\begin{patch}{PolynomialXmpPageEmpty27} 444\begin{paste}{PolynomialXmpPageEmpty27}{PolynomialXmpPagePatch27} 445\pastebutton{PolynomialXmpPageEmpty27}{\showpaste} 446\tab{5}\spadcommand{leadingMonomial p\free{p }} 447\end{paste}\end{patch} 448 449\begin{patch}{PolynomialXmpPagePatch28} 450\begin{paste}{PolynomialXmpPageFull28}{PolynomialXmpPageEmpty28} 451\pastebutton{PolynomialXmpPageFull28}{\hidepaste} 452\tab{5}\spadcommand{reductum p\free{p }} 453\indentrel{3}\begin{verbatim} 454 (28) (- 2 x y + x)z 455 Type: Polynomial(Integer) 456\end{verbatim} 457\indentrel{-3}\end{paste}\end{patch} 458 459\begin{patch}{PolynomialXmpPageEmpty28} 460\begin{paste}{PolynomialXmpPageEmpty28}{PolynomialXmpPagePatch28} 461\pastebutton{PolynomialXmpPageEmpty28}{\showpaste} 462\tab{5}\spadcommand{reductum p\free{p }} 463\end{paste}\end{patch} 464 465\begin{patch}{PolynomialXmpPagePatch29} 466\begin{paste}{PolynomialXmpPageFull29}{PolynomialXmpPageEmpty29} 467\pastebutton{PolynomialXmpPageFull29}{\hidepaste} 468\tab{5}\spadcommand{p - leadingMonomial p - reductum p\free{p }} 469\indentrel{3}\begin{verbatim} 470 (29) 0 471 Type: Polynomial(Integer) 472\end{verbatim} 473\indentrel{-3}\end{paste}\end{patch} 474 475\begin{patch}{PolynomialXmpPageEmpty29} 476\begin{paste}{PolynomialXmpPageEmpty29}{PolynomialXmpPagePatch29} 477\pastebutton{PolynomialXmpPageEmpty29}{\showpaste} 478\tab{5}\spadcommand{p - leadingMonomial p - reductum p\free{p }} 479\end{paste}\end{patch} 480 481\begin{patch}{PolynomialXmpPagePatch30} 482\begin{paste}{PolynomialXmpPageFull30}{PolynomialXmpPageEmpty30} 483\pastebutton{PolynomialXmpPageFull30}{\hidepaste} 484\tab{5}\spadcommand{leadingCoefficient p\free{p }} 485\indentrel{3}\begin{verbatim} 486 (30) 1 487 Type: PositiveInteger 488\end{verbatim} 489\indentrel{-3}\end{paste}\end{patch} 490 491\begin{patch}{PolynomialXmpPageEmpty30} 492\begin{paste}{PolynomialXmpPageEmpty30}{PolynomialXmpPagePatch30} 493\pastebutton{PolynomialXmpPageEmpty30}{\showpaste} 494\tab{5}\spadcommand{leadingCoefficient p\free{p }} 495\end{paste}\end{patch} 496 497\begin{patch}{PolynomialXmpPagePatch31} 498\begin{paste}{PolynomialXmpPageFull31}{PolynomialXmpPageEmpty31} 499\pastebutton{PolynomialXmpPageFull31}{\hidepaste} 500\tab{5}\spadcommand{p\free{p }} 501\indentrel{3}\begin{verbatim} 502 2 503 (31) (x y - 2 x y + x)z 504 Type: Polynomial(Integer) 505\end{verbatim} 506\indentrel{-3}\end{paste}\end{patch} 507 508\begin{patch}{PolynomialXmpPageEmpty31} 509\begin{paste}{PolynomialXmpPageEmpty31}{PolynomialXmpPagePatch31} 510\pastebutton{PolynomialXmpPageEmpty31}{\showpaste} 511\tab{5}\spadcommand{p\free{p }} 512\end{paste}\end{patch} 513 514\begin{patch}{PolynomialXmpPagePatch32} 515\begin{paste}{PolynomialXmpPageFull32}{PolynomialXmpPageEmpty32} 516\pastebutton{PolynomialXmpPageFull32}{\hidepaste} 517\tab{5}\spadcommand{eval(p,x,w)\free{p }} 518\indentrel{3}\begin{verbatim} 519 2 520 (32) (w y - 2 w y + w)z 521 Type: Polynomial(Integer) 522\end{verbatim} 523\indentrel{-3}\end{paste}\end{patch} 524 525\begin{patch}{PolynomialXmpPageEmpty32} 526\begin{paste}{PolynomialXmpPageEmpty32}{PolynomialXmpPagePatch32} 527\pastebutton{PolynomialXmpPageEmpty32}{\showpaste} 528\tab{5}\spadcommand{eval(p,x,w)\free{p }} 529\end{paste}\end{patch} 530 531\begin{patch}{PolynomialXmpPagePatch33} 532\begin{paste}{PolynomialXmpPageFull33}{PolynomialXmpPageEmpty33} 533\pastebutton{PolynomialXmpPageFull33}{\hidepaste} 534\tab{5}\spadcommand{eval(p,x,1)\free{p }} 535\indentrel{3}\begin{verbatim} 536 2 537 (33) (y - 2 y + 1)z 538 Type: Polynomial(Integer) 539\end{verbatim} 540\indentrel{-3}\end{paste}\end{patch} 541 542\begin{patch}{PolynomialXmpPageEmpty33} 543\begin{paste}{PolynomialXmpPageEmpty33}{PolynomialXmpPagePatch33} 544\pastebutton{PolynomialXmpPageEmpty33}{\showpaste} 545\tab{5}\spadcommand{eval(p,x,1)\free{p }} 546\end{paste}\end{patch} 547 548\begin{patch}{PolynomialXmpPagePatch34} 549\begin{paste}{PolynomialXmpPageFull34}{PolynomialXmpPageEmpty34} 550\pastebutton{PolynomialXmpPageFull34}{\hidepaste} 551\tab{5}\spadcommand{eval(p,x,y^2 - 1)\free{p }} 552\indentrel{3}\begin{verbatim} 553 4 3 554 (34) (y - 2 y + 2 y - 1)z 555 Type: Polynomial(Integer) 556\end{verbatim} 557\indentrel{-3}\end{paste}\end{patch} 558 559\begin{patch}{PolynomialXmpPageEmpty34} 560\begin{paste}{PolynomialXmpPageEmpty34}{PolynomialXmpPagePatch34} 561\pastebutton{PolynomialXmpPageEmpty34}{\showpaste} 562\tab{5}\spadcommand{eval(p,x,y^2 - 1)\free{p }} 563\end{paste}\end{patch} 564 565\begin{patch}{PolynomialXmpPagePatch35} 566\begin{paste}{PolynomialXmpPageFull35}{PolynomialXmpPageEmpty35} 567\pastebutton{PolynomialXmpPageFull35}{\hidepaste} 568\tab{5}\spadcommand{D(p,x)\free{p }} 569\indentrel{3}\begin{verbatim} 570 2 571 (35) (y - 2 y + 1)z 572 Type: Polynomial(Integer) 573\end{verbatim} 574\indentrel{-3}\end{paste}\end{patch} 575 576\begin{patch}{PolynomialXmpPageEmpty35} 577\begin{paste}{PolynomialXmpPageEmpty35}{PolynomialXmpPagePatch35} 578\pastebutton{PolynomialXmpPageEmpty35}{\showpaste} 579\tab{5}\spadcommand{D(p,x)\free{p }} 580\end{paste}\end{patch} 581 582\begin{patch}{PolynomialXmpPagePatch36} 583\begin{paste}{PolynomialXmpPageFull36}{PolynomialXmpPageEmpty36} 584\pastebutton{PolynomialXmpPageFull36}{\hidepaste} 585\tab{5}\spadcommand{D(p,y)\free{p }} 586\indentrel{3}\begin{verbatim} 587 (36) (2 x y - 2 x)z 588 Type: Polynomial(Integer) 589\end{verbatim} 590\indentrel{-3}\end{paste}\end{patch} 591 592\begin{patch}{PolynomialXmpPageEmpty36} 593\begin{paste}{PolynomialXmpPageEmpty36}{PolynomialXmpPagePatch36} 594\pastebutton{PolynomialXmpPageEmpty36}{\showpaste} 595\tab{5}\spadcommand{D(p,y)\free{p }} 596\end{paste}\end{patch} 597 598\begin{patch}{PolynomialXmpPagePatch37} 599\begin{paste}{PolynomialXmpPageFull37}{PolynomialXmpPageEmpty37} 600\pastebutton{PolynomialXmpPageFull37}{\hidepaste} 601\tab{5}\spadcommand{D(p,z)\free{p }} 602\indentrel{3}\begin{verbatim} 603 2 604 (37) x y - 2 x y + x 605 Type: Polynomial(Integer) 606\end{verbatim} 607\indentrel{-3}\end{paste}\end{patch} 608 609\begin{patch}{PolynomialXmpPageEmpty37} 610\begin{paste}{PolynomialXmpPageEmpty37}{PolynomialXmpPagePatch37} 611\pastebutton{PolynomialXmpPageEmpty37}{\showpaste} 612\tab{5}\spadcommand{D(p,z)\free{p }} 613\end{paste}\end{patch} 614 615\begin{patch}{PolynomialXmpPagePatch38} 616\begin{paste}{PolynomialXmpPageFull38}{PolynomialXmpPageEmpty38} 617\pastebutton{PolynomialXmpPageFull38}{\hidepaste} 618\tab{5}\spadcommand{integrate(p,y)\free{p }} 619\indentrel{3}\begin{verbatim} 620 1 3 2 621 (38) (- x y - x y + x y)z 622 3 623 Type: Polynomial(Fraction(Integer)) 624\end{verbatim} 625\indentrel{-3}\end{paste}\end{patch} 626 627\begin{patch}{PolynomialXmpPageEmpty38} 628\begin{paste}{PolynomialXmpPageEmpty38}{PolynomialXmpPagePatch38} 629\pastebutton{PolynomialXmpPageEmpty38}{\showpaste} 630\tab{5}\spadcommand{integrate(p,y)\free{p }} 631\end{paste}\end{patch} 632 633\begin{patch}{PolynomialXmpPagePatch39} 634\begin{paste}{PolynomialXmpPageFull39}{PolynomialXmpPageEmpty39} 635\pastebutton{PolynomialXmpPageFull39}{\hidepaste} 636\tab{5}\spadcommand{qr := monicDivide(p,x+1,x)\free{p }\bound{qr }} 637\indentrel{3}\begin{verbatim} 638 (39) 639 2 640 [quotient = (y - 2 y + 1)z, 641 2 642 remainder = (- y + 2 y - 1)z] 643Type: Record(quotient: Polynomial(Integer),remainder: Polynomial(Integer)) 644\end{verbatim} 645\indentrel{-3}\end{paste}\end{patch} 646 647\begin{patch}{PolynomialXmpPageEmpty39} 648\begin{paste}{PolynomialXmpPageEmpty39}{PolynomialXmpPagePatch39} 649\pastebutton{PolynomialXmpPageEmpty39}{\showpaste} 650\tab{5}\spadcommand{qr := monicDivide(p,x+1,x)\free{p }\bound{qr }} 651\end{paste}\end{patch} 652 653\begin{patch}{PolynomialXmpPagePatch40} 654\begin{paste}{PolynomialXmpPageFull40}{PolynomialXmpPageEmpty40} 655\pastebutton{PolynomialXmpPageFull40}{\hidepaste} 656\tab{5}\spadcommand{qr.remainder\free{qr }} 657\indentrel{3}\begin{verbatim} 658 2 659 (40) (- y + 2 y - 1)z 660 Type: Polynomial(Integer) 661\end{verbatim} 662\indentrel{-3}\end{paste}\end{patch} 663 664\begin{patch}{PolynomialXmpPageEmpty40} 665\begin{paste}{PolynomialXmpPageEmpty40}{PolynomialXmpPagePatch40} 666\pastebutton{PolynomialXmpPageEmpty40}{\showpaste} 667\tab{5}\spadcommand{qr.remainder\free{qr }} 668\end{paste}\end{patch} 669 670\begin{patch}{PolynomialXmpPagePatch41} 671\begin{paste}{PolynomialXmpPageFull41}{PolynomialXmpPageEmpty41} 672\pastebutton{PolynomialXmpPageFull41}{\hidepaste} 673\tab{5}\spadcommand{p - ((x+1) * qr.quotient + qr.remainder)\free{p }\free{qr }} 674\indentrel{3}\begin{verbatim} 675 (41) 0 676 Type: Polynomial(Integer) 677\end{verbatim} 678\indentrel{-3}\end{paste}\end{patch} 679 680\begin{patch}{PolynomialXmpPageEmpty41} 681\begin{paste}{PolynomialXmpPageEmpty41}{PolynomialXmpPagePatch41} 682\pastebutton{PolynomialXmpPageEmpty41}{\showpaste} 683\tab{5}\spadcommand{p - ((x+1) * qr.quotient + qr.remainder)\free{p }\free{qr }} 684\end{paste}\end{patch} 685 686\begin{patch}{PolynomialXmpPagePatch42} 687\begin{paste}{PolynomialXmpPageFull42}{PolynomialXmpPageEmpty42} 688\pastebutton{PolynomialXmpPageFull42}{\hidepaste} 689\tab{5}\spadcommand{p/q\free{p }\free{q }} 690\indentrel{3}\begin{verbatim} 691 (y - 1)z 692 (42) -------- 693 z + 5 694 Type: Fraction(Polynomial(Integer)) 695\end{verbatim} 696\indentrel{-3}\end{paste}\end{patch} 697 698\begin{patch}{PolynomialXmpPageEmpty42} 699\begin{paste}{PolynomialXmpPageEmpty42}{PolynomialXmpPagePatch42} 700\pastebutton{PolynomialXmpPageEmpty42}{\showpaste} 701\tab{5}\spadcommand{p/q\free{p }\free{q }} 702\end{paste}\end{patch} 703 704\begin{patch}{PolynomialXmpPagePatch43} 705\begin{paste}{PolynomialXmpPageFull43}{PolynomialXmpPageEmpty43} 706\pastebutton{PolynomialXmpPageFull43}{\hidepaste} 707\tab{5}\spadcommand{(2/3) * x^2 - y + 4/5\bound{prev1 }} 708\indentrel{3}\begin{verbatim} 709 2 2 4 710 (43) - y + - x + - 711 3 5 712 Type: Polynomial(Fraction(Integer)) 713\end{verbatim} 714\indentrel{-3}\end{paste}\end{patch} 715 716\begin{patch}{PolynomialXmpPageEmpty43} 717\begin{paste}{PolynomialXmpPageEmpty43}{PolynomialXmpPagePatch43} 718\pastebutton{PolynomialXmpPageEmpty43}{\showpaste} 719\tab{5}\spadcommand{(2/3) * x^2 - y + 4/5\bound{prev1 }} 720\end{paste}\end{patch} 721 722\begin{patch}{PolynomialXmpPagePatch44} 723\begin{paste}{PolynomialXmpPageFull44}{PolynomialXmpPageEmpty44} 724\pastebutton{PolynomialXmpPageFull44}{\hidepaste} 725\tab{5}\spadcommand{\% :: FRAC POLY INT\free{prev1 }\bound{prev2 }} 726\indentrel{3}\begin{verbatim} 727 2 728 - 15 y + 10 x + 12 729 (44) ------------------- 730 15 731 Type: Fraction(Polynomial(Integer)) 732\end{verbatim} 733\indentrel{-3}\end{paste}\end{patch} 734 735\begin{patch}{PolynomialXmpPageEmpty44} 736\begin{paste}{PolynomialXmpPageEmpty44}{PolynomialXmpPagePatch44} 737\pastebutton{PolynomialXmpPageEmpty44}{\showpaste} 738\tab{5}\spadcommand{\% :: FRAC POLY INT\free{prev1 }\bound{prev2 }} 739\end{paste}\end{patch} 740 741\begin{patch}{PolynomialXmpPagePatch45} 742\begin{paste}{PolynomialXmpPageFull45}{PolynomialXmpPageEmpty45} 743\pastebutton{PolynomialXmpPageFull45}{\hidepaste} 744\tab{5}\spadcommand{\% :: POLY FRAC INT\free{prev2 }\bound{prev3 }} 745\indentrel{3}\begin{verbatim} 746 2 2 4 747 (45) - y + - x + - 748 3 5 749 Type: Polynomial(Fraction(Integer)) 750\end{verbatim} 751\indentrel{-3}\end{paste}\end{patch} 752 753\begin{patch}{PolynomialXmpPageEmpty45} 754\begin{paste}{PolynomialXmpPageEmpty45}{PolynomialXmpPagePatch45} 755\pastebutton{PolynomialXmpPageEmpty45}{\showpaste} 756\tab{5}\spadcommand{\% :: POLY FRAC INT\free{prev2 }\bound{prev3 }} 757\end{paste}\end{patch} 758 759\begin{patch}{PolynomialXmpPagePatch46} 760\begin{paste}{PolynomialXmpPageFull46}{PolynomialXmpPageEmpty46} 761\pastebutton{PolynomialXmpPageFull46}{\hidepaste} 762\tab{5}\spadcommand{map(numeric,\%)\free{prev3 }} 763\indentrel{3}\begin{verbatim} 764 2 765 (46) - 1.0 y + 0.6666666666_6666666667 x + 0.8 766 Type: Polynomial(Float) 767\end{verbatim} 768\indentrel{-3}\end{paste}\end{patch} 769 770\begin{patch}{PolynomialXmpPageEmpty46} 771\begin{paste}{PolynomialXmpPageEmpty46}{PolynomialXmpPagePatch46} 772\pastebutton{PolynomialXmpPageEmpty46}{\showpaste} 773\tab{5}\spadcommand{map(numeric,\%)\free{prev3 }} 774\end{paste}\end{patch} 775 776