Annotation of OpenXM/src/k097/Doc/complex.texi, Revision 1.5
1.5 ! takayama 1: %% $OpenXM: OpenXM/src/k097/Doc/complex.texi,v 1.4 2001/01/13 01:17:36 takayama Exp $
1.1 takayama 2:
3: /*&ja
4:
5: @node COMPLEX $BH!?t(B,,, Top
6: @chapter COMPLEX $BH!?t(B
7: @section $BH!?t0lMw(B
8: */
9: /*&en
10:
11: @node COMPLEX function,,, Top
12: @chapter COMPLEX function
13: @section A list of functions
14: */
15: /*&C
16: @menu
17: * Res_solv::
18: * Res_solv2::
19: * Kernel::
20: * Kernel2::
21: * Gb::
22: * Gb_h::
23: * Res_shiftMatrix::
24: @end menu
25:
26: */
27:
28: /*&ja
29: @c %%%%%%%%%%%%%%%%%%%% start of Res_solv %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
30: @menu
31: * Res_solv::
32: @end menu
33: @node Res_solv,,, COMPLEX $BH!?t(B
34: @subsection @code{Res_solv}
35: @findex Res_solv
36: @table @t
37: @item Res_solv(@var{m},@var{d})
1.3 takayama 38: :: $B0l<!ITDjJ}Dx<0(B u @var{m} =@var{d} $B$N2r$r$b$H$a$k(B.
1.1 takayama 39: @item Res_solv(@var{m},@var{d},@var{r})
40: :: $B0l<!ITDjJ}Dx<0(B u @var{m} =@var{d} $B$N2r$r$b$H$a$k(B. @var{r} $B$O(B ring.
41: @end table
42:
43: */
44: /*&en
45: @c %%%%%%%%%%%%%%%%%%%% start of Res_solv %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
46: @menu
47: * Res_solv::
48: @end menu
49: @node Res_solv,,, COMPLEX function
50: @subsection @code{Res_solv}
51: @findex Res_solv
52: @table @t
53: @item Res_solv(@var{m},@var{d})
54: :: Find a solution u of the linear indefinite equation u @var{m} =@var{d}.
55: @item Res_solv(@var{m},@var{d},@var{r})
56: :: Find a solution u of the linear indefinite equation u @var{m} =@var{d}.
57: @var{r} is a ring object.
58: @end table
59:
60: */
61:
62: /*&ja
63: @table @var
64: @item return
65: [c,r] $B$,La$jCM$N$H$-(B c/r $B$,2r(B u ($B2#%Y%/%H%k(B).
66: @item m
67: $B9TNs$^$?$O%Y%/%H%k(B
68: @item d
69: $B%Y%/%H%k(B $B$^$?$O%9%+%i!<(B
70: @end table
71:
72: @itemize @bullet
73: @item $B0l<!ITDjJ}Dx<0(B u @var{m} =@var{d} $B$N2r$r$b$H$a$k(B.
74: @item @var{m}, @var{d} $B$N=g$K4D$NDj5A$r8!:w$7(B, $B$=$N4D$HF1$8JQ?t=89g$r(B
75: $B$b$DHyJ,:nMQAG4D(B(graded reverse lexicographic order)$B$GITDjJ}Dx<0$r2r$/(B.
76: $B4D(B @var{r} $B$,$"$?$($i$l$F$$$k$H$-$O(B, @var{r} $B$HF1$8JQ?t=89g$r$b$D(B
77: $BHyJ,:nMQAG4D(B(graded reverse lexicographic order)$B$GITDjJ}Dx<0$r2r$/(B.
78: @item @var{m}, @var{d} $B$,Dj?t@.J,$N$H$-$O(B, $B4D(B @var{r} $B$rM?$($kI,MW$,$"$k(B.
79: (@var{m}, @var{d} $B$h$j4D$N>pJs$r$H$j$@$;$J$$$?$a(B).
80: @end itemize
81:
82: */
83: /*&en
84: @table @var
85: @item return
86: When [c,r] is the return value, c/r is the solution u.
87: @item m
88: Matrix or vector
89: @item d
90: Vector or scalar
91: @end table
92:
93: @itemize @bullet
94: @item Find a solution u of the linear indefinite equation u @var{m} =@var{d}.
95: @item It solves the linear indefinite equation in the ring of differential
96: operators (with graded reverse lexicographic order) of the same set
97: of variables of the ring to which @var{m} or @var{d} belongs.
98: When the ring @var{r} is given,
99: it solves the linear indefinite equation in the ring of differential
100: operators (with graded reverse lexicographic order) of the same set
101: of variables of the ring @var{r}.
102: @item When @var{m} and @var{d} consist of constants, a ring @var{r}
103: should be given.
104: @end itemize
105:
106: */
107:
108: /*&C
109: @example
110: In(16)= RingD("x,y");
111: In(17)= mm=[Dx,Dy,x];
112: In(18)= Res_solv(mm,1):
113: [ [ x , 0 , -Dx ] , -1 ]
114: @end example
115: */
116: /*&ja
117: $B$3$l$O(B -x*Dx + 0*Dy+Dx*x = 1 $B$G$"$k$3$H$r<($9(B.
118: */
119: /*&en
120: The output means that -x*Dx + 0*Dy+Dx*x = 1.
121: */
122:
123: /*&C
124: @example
125: In(4)=RingD("x");
126: m=[ [x*Dx+2, 0],[Dx+3,x^3],[3,x],[Dx*(x*Dx+3)-(x*Dx+2)*(x*Dx-4),0]];
127: d=[1,0];
128: Res_solv(m,d):
129:
130: [ [ x^2*Dx-x*Dx-4*x-1 , 0 , 0 , x ] , -2 ]
131: @end example
132: */
133: /*&ja
134: $B$3$l$O(B
135: -(1/2)*(x^2*Dx-x*Dx-4*x-1)*[x*Dx+2, 0]-(1/2)*[Dx*(x*Dx+3)-(x*Dx+2)*(x*Dx-4),0]
136: = [1,0]
137: $B$G$"$k$3$H$r<($9(B.
138: */
139: /*&en
140: The output implies that
141: -(1/2)*(x^2*Dx-x*Dx-4*x-1)*[x*Dx+2, 0]-(1/2)*[Dx*(x*Dx+3)-(x*Dx+2)*(x*Dx-4),0]
142: = [1,0]
143: */
144:
145: /*&C
146:
147: @example
148:
149: In(4)= r=RingD("x,y");
150: In(5)= Res_solv([[1,2],[3,4]],[5,0],r):
151: [ [ 10 , -5 ] , -1 ]
152:
153: @end example
154:
155:
156: */
157:
158:
159: /*&ja
160:
161: @table @t
162: @item $B;2>H(B
163: @code{Res_solv_h}, @code{Kernel}, @code{GetRing}, @code{SetRing}.
1.3 takayama 164: @item Files
165: @code{lib/restriction/complex.k}
1.1 takayama 166: @end table
167: @c %%%%%%%%%%%%%%%%%%%% end of Res_solv %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
168: */
169: /*&en
170:
171: @table @t
1.3 takayama 172: @item See also
1.1 takayama 173: @code{Res_solv_h}, @code{Kernel}, @code{GetRing}, @code{SetRing}.
1.3 takayama 174: @item Files
175: @code{lib/restriction/complex.k}
1.1 takayama 176: @end table
177: @c %%%%%%%%%%%%%%%%%%%% end of Res_solv %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
178: */
179:
180: /*&ja
181: @c %%%%%%%%%%%%%%%%%%%% start of Res_solv2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
182: @menu
183: * Res_solv2::
184: @end menu
185: @node Res_solv2,,, COMPLEX $BH!?t(B
186: @subsection @code{Res_solv2}
187: @findex Res_solv2
188: @table @t
189: @item Res_solv2(@var{m},@var{v},@var{j})
1.3 takayama 190: :: $B0l<!ITDjJ}Dx<0(B u @var{m} =@var{v} mod @var{j} $B$N2r$r$b$H$a$k(B.
1.1 takayama 191: @item Res_solv2(@var{m},@var{v},@var{j},@var{r})
192: :: $B0l<!ITDjJ}Dx<0(B u @var{m} =@var{v} mod @var{j} $B$N2r$r$b$H$a$k(B.
193: @var{r} $B$O(B ring.
194: @end table
195:
196: */
197: /*&en
198: @c %%%%%%%%%%%%%%%%%%%% start of Res_solv2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
199: @menu
200: * Res_solv2::
201: @end menu
202: @node Res_solv2,,, COMPLEX function
203: @subsection @code{Res_solv2}
204: @findex Res_solv2
205: @table @t
206: @item Res_solv2(@var{m},@var{v},@var{j})
207: :: Find a solution u of the linear indefinite equation u @var{m} =@var{v}
1.3 takayama 208: mod @var{j}.
1.1 takayama 209: @item Res_solv2(@var{m},@var{v},@var{j},@var{r})
210: :: Find a solution u of the linear indefinite equation u @var{m} =@var{v}
211: mod @var{j}.
212: @var{r} is a ring object.
213: @end table
214:
215: */
216:
217: /*&ja
218: @table @var
219: @item return
220: [c,r] $B$,La$jCM$N$H$-(B c/r $B$,2r(B u ($B2#%Y%/%H%k(B).
221: @item m
222: $B9TNs$^$?$O%Y%/%H%k(B
223: @item v j
224: $B%Y%/%H%k(B $B$^$?$O%9%+%i!<(B
225: @end table
226:
227: @itemize @bullet
228: @item $B0l<!ITDjJ}Dx<0(B u @var{m} =@var{v} mod @var{j} $B$N2r$r$b$H$a$k(B.
229: @item $B$3$l$O(B, @var{m} $B$r(B
230: @var{m} : D^p ---> D^q/@var{j}
231: $B$J$k:8(B D homomorphism $B$H$_$J$9$H$-(B,
232: @var{m}^(-1)(@var{v}) $B$r5a$a$k$3$H$KAjEv$9$k(B.
233: @item @var{m}, @var{v} $B$N=g$K4D$NDj5A$r8!:w$7(B, $B$=$N4D$HF1$8JQ?t=89g$r(B
234: $B$b$DHyJ,:nMQAG4D(B(graded reverse lexicographic order)$B$GITDjJ}Dx<0$r2r$/(B.
235: $B4D(B @var{r} $B$,$"$?$($i$l$F$$$k$H$-$O(B, @var{r} $B$HF1$8JQ?t=89g$r$b$D(B
236: $BHyJ,:nMQAG4D(B(graded reverse lexicographic order)$B$GITDjJ}Dx<0$r2r$/(B.
237: @item @var{m}, @var{v} $B$,Dj?t@.J,$N$H$-$O(B, $B4D(B @var{r} $B$rM?$($kI,MW$,$"$k(B.
238: (@var{m}, @var{v} $B$h$j4D$N>pJs$r$H$j$@$;$J$$$?$a(B).
239: @end itemize
240:
241: */
242: /*&en
243: @table @var
244: @item return
245: When [c,r] is the return value, c/r is the solution u.
246: @item m
247: Matrix or vector
248: @item v j
249: Vector or scalar
250: @end table
251:
252: @itemize @bullet
253: @item Find a solution u of the linear indefinite equation u @var{m} =@var{v}
254: mod @var{j}.
255: @item Let @var{m} be the left D-homomorphism
256: @var{m} : D^p ---> D^q/@var{j}.
257: The function returns an element in
258: @var{m}^(-1)(@var{v}).
259: @item It solves the linear indefinite equation in the ring of differential
260: operators (with graded reverse lexicographic order) of the same set
261: of variables of the ring to which @var{m} or @var{v} belongs.
262: When the ring @var{r} is given,
263: it solves the linear indefinite equation in the ring of differential
264: operators (with graded reverse lexicographic order) of the same set
265: of variables of the ring @var{r}.
266: @item When @var{m} and @var{v} consist of constants, a ring @var{r}
267: should be given.
268: @end itemize
269:
270: */
271:
272: /*&C
273: @example
274: In(28)= r=RingD("x,y");
275: In(29)= Res_solv2([x,y],[x^2+y^2],[x]):
276: [ [ 0 , y ] , 1 ]
277:
278: @end example
279: */
280: /*&ja
281: $B$3$l$O(B 0*x + y*y = x^2+y^2 mod x $B$G$"$k$3$H$r<($9(B.
282: */
283: /*&en
284: The output means that 0*x + y*y = x^2+y^2 mod x
285: */
286:
287: /*&C
288: @example
289:
290: In(32)= Res_solv2([x,y],[x^2+y^2],[],r):
291: [ [ x , y ] , 1 ]
292:
293:
294: @end example
295: */
296: /*&ja
297: $B$3$l$O(B
298: x*x + y*y = x^2+y^2
299: $B$G$"$k$3$H$r<($9(B.
300: */
301: /*&en
302: The output implies that
303: x*x + y*y = x^2+y^2.
304: */
305:
306:
307:
308: /*&ja
309:
310: @table @t
311: @item $B;2>H(B
312: @code{Res_solv2_h}, @code{Kernel2}, @code{GetRing}, @code{SetRing}.
1.3 takayama 313: @item Files
314: @code{lib/restriction/complex.k}
1.1 takayama 315: @end table
316: @c %%%%%%%%%%%%%%%%%%%% end of Res_solv2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
317: */
318: /*&en
319:
320: @table @t
1.3 takayama 321: @item See also
1.1 takayama 322: @code{Res_solv2_h}, @code{Kernel2}, @code{GetRing}, @code{SetRing}.
1.3 takayama 323: @item Files
324: @code{lib/restriction/complex.k}
1.1 takayama 325: @end table
326: @c %%%%%%%%%%%%%%%%%%%% end of Res_solv2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
327: */
328:
329:
330: /*&ja
331: @c %%%%%%%%%%%%%%%%%%%% start of Kernel %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
332: @c Kernel $B$O(B minimal.k $B$K$"$k$,(B complex.k $B$J$I$K0\F0$9$Y$-(B.
333: @menu
334: * Kernel::
335: @end menu
336: @node Kernel,,, COMPLEX $BH!?t(B
337: @subsection @code{Kernel}
338: @findex Kernel
339: @table @t
340: @item Kernel(@var{m})
341: :: $B0l<!ITDjJ}Dx<0(B u @var{m} =0 $B$N2r6u4V$N4pDl$r5a$a$k(B.
342: @item Kernel(@var{m},@var{r})
343: :: $B0l<!ITDjJ}Dx<0(B u @var{m} =0 $B$N2r6u4V$N4pDl$r5a$a$k(B. @var{r} $B$O(B ring.
344: @end table
345:
346: */
347: /*&en
348: @c %%%%%%%%%%%%%%%%%%%% start of Kernel %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
349: @menu
350: * Kernel::
351: @end menu
352: @node Kernel,,, COMPLEX function
353: @subsection @code{Kernel}
354: @findex Kernel
355: @table @t
356: @item Kernel(@var{m})
357: :: Find solution basis of the linear indefinite equation u @var{m} =0.
358: @item Kernel(@var{m},@var{r})
359: :: Find solution basis of the linear indefinite equation u @var{m} =0.
360: @var{r} is a ring object.
361: @end table
362:
363: */
364:
365: /*&ja
366: @table @var
367: @item return
368: $B%j%9%H(B
369: @item m
370: $B9TNs$^$?$O%Y%/%H%k(B
371: @end table
372:
373: @itemize @bullet
374: @item $B0l<!ITDjJ}Dx<0(B u @var{m} =0 $B$N2r6u4V$N4pDl$r5a$a$k(B.
375: @item $BLa$jCM$r(B k $B$H$9$k$H$-(B k[0] $B$,(B $B2r6u4V$N4pDl$N=89g(B.
376: k[1] $B$O(B [gb, backward transformation, syzygy without dehomogenization].
377: @item @var{m} $B$h$j4D$NDj5A$r8!:w$7(B, $B$=$N4D$G%+!<%M%k$r7W;;$9$k(B.
378: $B4D(B @var{r} $B$,$"$?$($i$l$F$$$k$H$-$O(B, @var{r} $B$G%+!<%M%k$r7W;;$9$k(B.
379: @item @var{m} $B$,Dj?t@.J,$N$H$-$O(B, $B4D(B @var{r} $B$rM?$($kI,MW$,$"$k(B.
380: (@var{m} $B$h$j4D$N>pJs$r$H$j$@$;$J$$$?$a(B).
381: @item BUG: Kernel $B$*$h$S(B Res_solv (syz, res-solv) $B$N$_$,(B, $B4D0z?t$K(B
382: $BBP1~$7$F$k(B. (2000, 12/29 $B8=:_(B).
383: @end itemize
384:
385: */
386: /*&en
387: @table @var
388: @item return
389: List
390: @item m
391: Matrix or vector
392: @end table
393:
394: @itemize @bullet
395: @item Find solution basis of the linear indefinite equation u @var{m} =0.
396: @item When the return value is k, k[0] is a set of generators of the kernel.
397: k[1] is [gb, backward transformation, syzygy without dehomogenization].
398: @item It finds the kernel in the ring
399: to which @var{m} belongs.
400: When the ring @var{r} is given,
401: it finds the kernel in the ring @var{r}.
402: @item When @var{m} consists of constants, a ring @var{r}
403: should be given.
404: @end itemize
405:
406: */
407:
408: /*&C
409: @example
410: In(16)= RingD("x,y");
411: In(17)= mm=[[Dx],[Dy],[x]];
412: In(18)= Pmat(Kernel(mm));
413: [
414: [
415: [ -x*Dx-2 , 0 , Dx^2 ]
416: [ -x*Dy , -1 , Dx*Dy ]
417: [ -x^2 , 0 , x*Dx-1 ]
418: ]
419: [
420: [
421: [ -1 ]
422: ]
423: [
424: [ x , 0 , -Dx ]
425: ]
426: [
427: [ -x*Dx-2 , 0 , Dx^2 ]
428: [ -x*Dy , -1 , Dx*Dy ]
429: [ -x^2 , 0 , x*Dx-1 ]
430: ]
431: ]
432: ]
433:
434: @end example
435: */
436:
437:
438: /*&C
439:
440: @example
441:
442: In(4)= r=RingD("x,y");
443: In(5)= k=Kernel([[1,2],[2,4]],r); k[0]:
444: [ [ 2 , -1 ] ]
445: @end example
446:
447:
448: */
449:
450:
451: /*&ja
452:
453: @table @t
454: @item $B;2>H(B
455: @code{Kernel_h}, @code{Res_solv}, @code{GetRing}, @code{SetRing}.
1.3 takayama 456: @item Files
457: @code{lib/restriction/complex.k}
1.1 takayama 458: @end table
459: @c %%%%%%%%%%%%%%%%%%%% end of Kernel %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
460: */
461: /*&en
462:
463: @table @t
1.3 takayama 464: @item See also
1.1 takayama 465: @code{Kernel_h}, @code{Res_solv}, @code{GetRing}, @code{SetRing}.
1.3 takayama 466: @item Files
467: @code{lib/restriction/complex.k}
1.1 takayama 468: @end table
469: @c %%%%%%%%%%%%%%%%%%%% end of Kernel %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
470: */
471: /*&ja
472: @c %%%%%%%%%%%%%%%%%%%% start of Kernel2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
473: @menu
474: * Kernel2::
475: @end menu
476: @node Kernel2,,, COMPLEX $BH!?t(B
477: @subsection @code{Kernel2}
478: @findex Kernel2
479: @table @t
480: @item Kernel2(@var{m},@var{j})
481: :: @var{m} : D^p ---> D^q/@var{j} $B$N(B Kernel $B$r5a$a$k(B.
482: @item Kernel2(@var{m},@var{j},@var{r})
483: :: @var{m} : D^p ---> D^q/@var{j} $B$N(B Kernel $B$r5a$a$k(B. @var{r} $B$O(B ring.
484: @end table
485:
486: */
487: /*&en
488: @c %%%%%%%%%%%%%%%%%%%% start of Kernel2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
489: @menu
490: * Kernel2::
491: @end menu
492: @node Kernel2,,, COMPLEX function
493: @subsection @code{Kernel2}
494: @findex Kernel2
495: @table @t
496: @item Kernel2(@var{m})
497: :: Get the kernel of @var{m} : D^p ---> D^q/@var{j}.
498: @item Kernel2(@var{m},@var{r})
499: :: Get the kernel of @var{m} : D^p ---> D^q/@var{j}.
500: @var{r} is a ring object.
501: @end table
502:
503: */
504:
505: /*&ja
506: @table @var
507: @item return
508: $B%j%9%H(B
509: @item m j
510: $B9TNs$^$?$O%Y%/%H%k(B
511: @end table
512:
513: @itemize @bullet
514: @item @var{m} : D^p ---> D^q/@var{j} $B$N(B Kernel $B$r5a$a$k(B.
515: @item D^p $B$O2#%Y%/%H%k$G$"$j(B, u $B$,(B D^p $B$N85$N$H$-(B,
516: u @var{m} $B$G<LA|$rDj5A$9$k(B.
517: @item @var{m} $B$h$j4D$NDj5A$r8!:w$7(B, $B$=$N4D$G%+!<%M%k$r7W;;$9$k(B.
518: $B4D(B @var{r} $B$,$"$?$($i$l$F$$$k$H$-$O(B, @var{r} $B$G%+!<%M%k$r7W;;$9$k(B.
519: @item @var{m} $B$,Dj?t@.J,$N$H$-$O(B, $B4D(B @var{r} $B$rM?$($kI,MW$,$"$k(B.
520: (@var{m} $B$h$j4D$N>pJs$r$H$j$@$;$J$$$?$a(B).
521: @end itemize
522:
523: */
524: /*&en
525: @table @var
526: @item return
527: List
528: @item m j
529: Matrix or vector
530: @end table
531:
532: @itemize @bullet
533: @item Get a set of generators of the the kernel
534: of @var{m} : D^p ---> D^q/@var{j}.
535: @item D^p is a set of row vectors. When u is an element of D^p,
536: define the map from D^p to D^q/@var{j} by u @var{m}.
537: @item It finds the kernel in the ring
538: to which @var{m} belongs.
539: When the ring @var{r} is given,
540: it finds the kernel in the ring @var{r}.
541: @item When @var{m} consists of constants, a ring @var{r}
542: should be given.
543: @end itemize
544:
545: */
546:
547: /*&C
548: @example
549: In(27)= r=RingD("x,y");
550: In(28)= Kernel2([[x,y],[x^2,x*y]],[]):
551: [ [ -x , 1 ] ]
552: In(29)=Kernel2([[x,y],[x^2,x*y]],[[x,y]]):
553: [ [ 1 , 0 ] , [ 0 , 1 ] ]
554:
555: In(41)=Kernel2([0],[0],r):
556: [ [ 1 ] , [ 0 ] ]
557: In(42)=Kernel2([[0,0],[0,0]],[[0,0]],r):
558: [ [ 1 , 0 ] , [ 0 , 1 ] , [ 0 , 0 ] ]
559: In(43)=Kernel2([[0,0,0],[0,0,0]],[],r):
560: [ [ 1 , 0 ] , [ 0 , 1 ] ]
561:
562: @end example
563: */
564:
565:
566: /*&ja
567:
568: @table @t
569: @item $B;2>H(B
570: @code{Kernel2_h}, @code{Res_solv2}, @code{GetRing}, @code{SetRing},
571: @code{Kernel}.
1.3 takayama 572: @item Files
573: @code{lib/restriction/complex.k}
1.1 takayama 574: @end table
575: @c %%%%%%%%%%%%%%%%%%%% end of Kernel2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
576: */
577: /*&en
578:
579: @table @t
1.3 takayama 580: @item See also
1.1 takayama 581: @code{Kernel2_h}, @code{Res_solv2}, @code{GetRing}, @code{SetRing},
582: @code{Kernel}
1.3 takayama 583: @item Files
584: @code{lib/restriction/complex.k}
1.1 takayama 585: @end table
586: @c %%%%%%%%%%%%%%%%%%%% end of Kernel2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
587: */
588:
589: /*&ja
590: @c %%%%%%%%%%%%%%%%%%%% start of Gb %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
591: @menu
592: * Gb::
593: @end menu
594: @node Gb,,, COMPLEX $BH!?t(B
595: @node Gb_h,,, COMPLEX $BH!?t(B
596: @subsection @code{Gb}
597: @findex Gb
598: @findex Gb_h
599: @table @t
600: @item Gb(@var{f})
601: :: @var{f} $B$N%0%l%V%J4pDl$r$b$H$a$k(B.
602: @item Gb(@var{f},@var{r})
603: :: @var{f} $B$N%0%l%V%J4pDl$r$b$H$a$k(B. @var{r} $B$O(B ring.
604: @item Gb_h(@var{f})
605: :: @var{f} $B$N%0%l%V%J4pDl$r$b$H$a$k(B.
606: @item Gb_h(@var{f},@var{r})
607: :: @var{f} $B$N%0%l%V%J4pDl$r$b$H$a$k(B. @var{r} $B$O(B ring.
608: @end table
609:
610: */
611: /*&en
612: @c %%%%%%%%%%%%%%%%%%%% start of Gb %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
613: @menu
614: * Gb::
615: @end menu
616: @node Gb,,, COMPLEX function
617: @node Gb_h,,, COMPLEX function
618: @subsection @code{Gb}
619: @findex Gb
620: @table @t
621: @item Gb(@var{f})
622: :: It computes the Grobner basis of @var{f}.
623: @item Gb(@var{m},@var{r})
624: :: It computes the Grobner basis of @var{f}.
625: @var{r} is a ring object.
626: @item Gb_h(@var{f})
627: :: It computes the Grobner basis of @var{f}.
628: @item Gb_h(@var{m},@var{r})
629: :: It computes the Grobner basis of @var{f}.
630: @var{r} is a ring object.
631: @end table
632:
633: */
634:
635: /*&ja
636: @table @var
637: @item return
638: $B%j%9%H(B
639: @item f
640: $B9TNs$^$?$O%Y%/%H%k(B
641: @end table
642:
643: @itemize @bullet
644: @item @var{f} $B$N%0%l%V%J4pDl$r$b$H$a$k(B.
645: @item _h $BIU$-$N>l9g$O(B, $BF1<!%o%$%kBe?t$G7W;;$r$*$3$J$&(B.
646: @item $BLa$jCM$r(B k $B$H$9$k$H$-(B k[0] $B$,(B $B%0%l%V%J4pDl(B.
647: $B4D$,(B weight vector $BIU$-$GDj5A$5$l$?$H$-$O(B,
648: k[1] $B$O(B initial ideal $B$^$?$O(B initial module.
649: @item @var{m} $B$h$j4D$NDj5A$r8!:w$7(B, $B$=$N4D$G%0%l%V%J4pDl$r7W;;$9$k(B.
650: $B4D(B @var{r} $B$,$"$?$($i$l$F$$$k$H$-$O(B, @var{r} $B$G%0%l%V%J4pDl$r7W;;$9$k(B.
651: @item @var{m} $B$,Dj?t@.J,$N$H$-$O(B, $B4D(B @var{r} $B$rM?$($kI,MW$,$"$k(B.
652: (@var{m} $B$h$j4D$N>pJs$r$H$j$@$;$J$$$?$a(B).
653: @end itemize
654:
655: */
656: /*&en
657: @table @var
658: @item return
659: List
660: @item f
661: Matrix or vector
662: @end table
663:
664: @itemize @bullet
665: @item Compute the Grobner basis of @var{f}.
666: @item Functions with _h computes Grobner bases in the homogenized Weyl
667: algebra.
668: @item When the return value is k, k[0] is the Grobner basis.
669: k[1] is the initial ideal or the initial module of @var{f},
670: when the ring is defined with a weight vector.
671: @item It computes the Grobner basis in the ring
672: to which @var{f} belongs.
673: When the ring @var{r} is given,
674: it computes the Grobner basis in the ring @var{r}.
675: @item When @var{f} consists of constants, a ring @var{r}
676: should be given.
677: @end itemize
678:
679: */
680:
681: /*&C
682: @example
683: In(5)= r=RingD("x,y");
684: In(6)= m=[[x^2+y^2-1],[x*y-1]];
685: In(7)= Gb(m):
686: [ [ [ x^2+y^2-1 ] , [ x*y-1 ] , [ y^3+x-y ] ] ,
687: [ [ x^2+y^2-1 ] , [ x*y-1 ] , [ y^3+x-y ] ] ]
688:
689: In(11)= RingD("x,y",[["x",1]]);
690: In(12)= r=RingD("x,y",[["x",1]]);
691: In(13)= Gb(m,r):
692: [ [ [ x+y^3-y ] , [ -y^4+y^2-1 ] ] ,
693: [ [ x ] , [ -y^4+y^2-1 ] ] ]
694:
695: @end example
696: */
697:
698:
699: /*&ja
700:
701: @table @t
702: @item $B;2>H(B
703: @code{Gb_h}, @code{Kernel}, @code{Res_solv}, @code{RingD},
704: @code{GetRing}, @code{SetRing}.
1.3 takayama 705: @item Files
706: @code{lib/restriction/complex.k}
1.1 takayama 707: @end table
708: @c %%%%%%%%%%%%%%%%%%%% end of Gb %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
709: */
710: /*&en
711:
712: @table @t
1.3 takayama 713: @item See also
1.1 takayama 714: @code{Gb_h}, @code{Kernel}, @code{Res_solv}, @code{RingD},
715: @code{GetRing}, @code{SetRing}.
1.3 takayama 716: @item Files
717: @code{lib/restriction/complex.k}
1.1 takayama 718: @end table
719: @c %%%%%%%%%%%%%%%%%%%% end of Gb %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
720: */
721:
722: /*&ja
723: @c %%%%%%%%%%%%%%%%%%%% start of Res_shiftMatrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
724: @menu
725: * Res_shiftMatrix::
726: @end menu
727: @node Res_shiftMatrix,,, COMPLEX $BH!?t(B
728: @subsection @code{Res_shiftMatrix}
729: @findex Res_shiftMatrix
730: @table @t
731: @item Res_shiftMatrix(@var{m},@var{v})
732: :: Degree shift $B%Y%/%H%k(B @var{m} $B$KBP1~$9$k9TNs$r:n$k(B.
733: @item Res_shiftMatrix(@var{f},@var{v},@var{r})
734: :: Degree shift $B%Y%/%H%k(B @var{m} $B$KBP1~$9$k9TNs$r:n$k(B. @var{r} $B$O(B ring.
735: @end table
736:
737: */
738: /*&en
739: @c %%%%%%%%%%%%%%%%%%%% start of Res_shiftMatrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
740: @menu
741: * Res_shiftMatrix::
742: @end menu
743: @node Res_shiftMatrix,,, COMPLEX function
744: @subsection @code{Res_shiftMatrix}
745: @findex Res_shiftMatrix
746: @table @t
747: @item Res_shiftMatrix(@var{m},@var{v})
748: :: Generate a matrix associated to a degree shift vector @var{m}
749: @item Res_shiftMatrix(@var{m},@var{v},@var{r})
750: :: Generate a matrix associated to a degree shift vector @var{m}
751: @var{r} is a ring object.
752: @end table
753:
754: */
755:
756: /*&ja
757: @table @var
758: @item return
759: $B9TNs(B.
760: @item m
761: $B%Y%/%H%k(B
762: @item v
763: $BB?9`<0JQ?t$^$?$OJ8;zNs(B
764: @end table
765:
766: @itemize @bullet
767: @item diag(@var{v}^(@var{m}1), ..., @var{v}^(@var{m}n))
768: $B$J$k(B n $B!_(B n $B9TNs$rLa$9(B.
769: @end itemize
770:
771: */
772: /*&en
773: @table @var
774: @item return
775: Matrix
776: @item m
777: Vector
778: @item v
779: $BB?9`<0JQ?t$^$?$OJ8;zNs(B
780: @end table
781:
782: @itemize @bullet
783: @item Returns n by n matrix
784: diag(@var{v}^(@var{m}1), ..., @var{v}^(@var{m}n))
785: @end itemize
786:
787: */
788:
789: /*&C
790: @example
791: In(5)= r=RingD("x,y");
792: In(6)= Res_shiftMatrix([-1,0,3],x):
793: [ [ x^(-1) , 0 , 0 ] , [ 0 , 1 , 0 ] , [ 0 , 0 , x^3 ] ]
794:
795: @end example
796: */
797:
798: /*&C
799: @example
800: In(9)= rrr = RingD("t,x,y",[["t",1,"x",-1,"y",-1,"Dx",1,"Dy",1]]);
801: In(10)= m=[Dx-(x*Dx+y*Dy+2),Dy-(x*Dx+y*Dy+2)];
802: In(12)= m=Gb(m);
803: In(13)= k = Kernel_h(m[0]);
804: In(14)= Pmat(k[0]);
805: [
806: [ -Dy+3*h , Dx-3*h , 1 ]
807: [ -x*Dx+x*Dy-y*Dy-3*x*h , y*Dy+3*x*h , h-x ]
808: ]
809:
810: In(15)=Pmat(m[0]);
811: [ Dx*h-x*Dx-y*Dy-2*h^2 , Dy*h-x*Dx-y*Dy-2*h^2 ,
812: x*Dx^2-x*Dx*Dy+y*Dx*Dy-y*Dy^2 ]
813:
814: In(18)=k2 = Gb_h(k[0]*Res_shiftMatrix([1,1,1],t));
815: In(19)=Pmat(Substitute(k2[0],t,1));
816: [
817: [ -Dy+3*h , Dx-3*h , 1 ]
818: [ -x*Dx+x*Dy-y*Dy-3*x*h , y*Dy+3*x*h , h-x ]
819: ]
820:
821:
822: @end example
823: */
824:
825:
826: /*&ja
827:
828: @table @t
829: @item $B;2>H(B
830: @code{Gb}, (m,(u,v))-$B%0%l%V%J4pDl(B
1.3 takayama 831: @item Files
832: @code{lib/restriction/complex.k}
1.1 takayama 833: @end table
834: @c %%%%%%%%%%%%%%%%%%%% end of Res_shiftMatrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
835: */
836: /*&en
837:
838: @table @t
1.3 takayama 839: @item See also
1.1 takayama 840: @code{Gb}, (m,(u,v))-Grobner basis
1.3 takayama 841: @item Files
842: @code{lib/restriction/complex.k}
1.1 takayama 843: @end table
844: @c %%%%%%%%%%%%%%%%%%%% end of Res_shiftMatrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
845: */
846:
847:
848: @c -------------- Primitive $B4pK\(B ----------------------
849:
850: /*&ja
851:
852: @node $B4pK\(B $BH!?t(B,,, Top
853: @chapter $B4pK\(B $BH!?t(B
854: @section $BH!?t0lMw(B
855: */
856: /*&en
857:
858: @node Primitive function,,, Top
859: @chapter Primitive function
860: @section A list of functions
861: */
862: /*&C
863: @menu
1.2 takayama 864: * ChangeRing::
1.1 takayama 865: * Intersection::
866: * Getxvars::
867: * Firstn::
868: @end menu
869: */
870:
871: /*&ja
1.2 takayama 872: @c %%%%%%%%%%%%%%%%%%%% start ChangeRing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
873: @node ChangeRing,,, $B4pK\(B $BH!?t(B
874: @subsection @code{ChangeRing}
875: @findex ChangeRing
1.1 takayama 876: @table @t
1.2 takayama 877: @item ChangeRing(@var{f})
878: :: ChangeRing $B$O(B @var{f} $B$NMWAG$,B0$9$k4D$r(B current ring $B$K$9$k(B.
1.1 takayama 879: @end table
880:
881: @table @var
882: @item return
883: true $B$+(B false
884: @item f $B%j%9%H(B
885: @end table
886:
887: @example
888: RingD("x,y");
889: f=[x+y,0];
890: RingD("p,q,r");
1.2 takayama 891: ChangeRing(f);
1.1 takayama 892: @end example
893:
894: @table @t
1.3 takayama 895: @item Files
896: @code{lib/restriction/complex.k}
1.1 takayama 897: @end table
1.3 takayama 898: @c %%%%%%%%%%%%%%%%%%%% end of ChangeRing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1.1 takayama 899: */
900:
901:
902: /*&ja
903: @c %%%%%%%%%%%%%%%%%%%% start of Intersection %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
904: @menu
905: * Intersection::
906: @end menu
907: @node Intersection,,, $B4pK\(B $BH!?t(B
908: @subsection @code{Intersection}
909: @findex Intersection
910: @table @t
911: @item Intersection(@var{i},@var{j})
912: :: $B%$%G%"%k(B @var{i} $B$H(B @var{j} $B$N8r$o$j$r5a$a$k(B.
913: @item Intersection(@var{i},@var{j},@var{r})
914: :: $B%$%G%"%k(B @var{i} $B$H(B @var{j} $B$N8r$o$j$r5a$a$k(B. $B7W;;$r4D(B @var{r}
915: $B$G$*$3$J$&(B.
916: @end table
917:
918: @table @var
919: @item return
920: $B%j%9%H$G$"$?$($i$l$?%$%G%"%k$^$?$O<+M32C72$NItJ,2C72(B
921: @item i j
922: $B%$%G%"%k$^$?$O<+M32C72$NItJ,2C72(B
923: @item r
924: $B4D(B
925: @end table
926:
927: @itemize @bullet
928: @item :: $B%$%G%"%k(B @var{i} $B$H(B @var{j} $B$N8r$o$j$r5a$a$k(B.
929: @end itemize
930:
931: @example
932: In(16)= RingD("x,y");
933: In(17)= mm=[ [x,0],[0,y] ]; nn = [ [x^2,0],[0,y^3]];
934: In(19)= Intersection(mm,nn):
935: In(33)=Intersection(mm,nn):
936: [ [ -x^2 , 0 ] , [ 0 , -y^3 ] ]
937: @end example
938:
939: @table @t
940: @item $B;2>H(B
1.3 takayama 941: @item Files
942: @code{lib/restriction/complex.k}
1.1 takayama 943: @end table
944: @c %%%%%%%%%%%%%%%%%%%% end of Intersection %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
945: */
946:
947: /*&ja
948: @c %%%%%%%%%%%%%%%%%%%% start of Getxvars %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
949: @menu
950: * Getxvars::
951: @end menu
952: @node Getxvars,,, $B4pK\(B $BH!?t(B
953: @subsection @code{Getxvars}
954: @findex Getxvars
955: @table @t
956: @item Getxvars()
957: :: x $BJQ?t$rLa$9(B
958: @end table
959:
960: */
961: /*&en
962: @c %%%%%%%%%%%%%%%%%%%% start of Getxvars %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
963: @menu
964: * Getxvars::
965: @end menu
966: @node Getxvars,,, Primitive function
967: @subsection @code{Getxvars}
968: @findex Getxvars
969: @table @t
970: @item Getxvars()
971: :: Return x variables
972: @end table
973:
974: */
975:
976: /*&ja
977: @table @var
978: @item return
979: [x_list, x_str] x_list $B$O(B x $BJQ?t$N%j%9%H(B, x_str $B$O(B x $BJQ?t$r(B , $B$G6h@Z$C$?J8;zNs(B.
980: @end table
981:
982: */
983: /*&en
984: @table @var
985: @item return
986: [x_list, x_str] x_list is a list of x variables, x_str is a string consisting
987: of x variables separated by commas.
988: @end table
989:
990:
991: */
992:
993: /*&C
994: @example
995: In(4)=RingD("x,y");
996: In(5)=Getxvars():
997: [ [ y , x ] , y,x, ]
998: @end example
999:
1000: */
1001:
1002:
1003: /*&ja
1004:
1005: @table @t
1006: @item $B;2>H(B
1.3 takayama 1007: @item Files
1008: @code{lib/restriction/complex.k}
1.1 takayama 1009: @end table
1010: @c %%%%%%%%%%%%%%%%%%%% end of Getxvars %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1011: */
1012: /*&en
1013:
1014: @table @t
1.3 takayama 1015: @item See also
1016: @item Files
1017: @code{lib/restriction/complex.k}
1.1 takayama 1018: @end table
1019: @c %%%%%%%%%%%%%%%%%%%% end of Getxvars %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1020: */
1021:
1022: /*&ja
1023: @c %%%%%%%%%%%%%%%%%%%% start of Firstn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1024: @menu
1025: * Firstn::
1026: @end menu
1027: @node Firstn,,, $B4pK\(B $BH!?t(B
1028: @subsection @code{Firstn}
1029: @findex Firstn
1030: @table @t
1031: @item Firstn(@var{m},@var{n})
1032: :: @var{m} $B$N:G=i$N(B @var{n} $B8D$r$H$j$@$9(B.
1033: @end table
1034:
1035: */
1036: /*&en
1037: @c %%%%%%%%%%%%%%%%%%%% start of Firstn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1038: @menu
1039: * Firstn::
1040: @end menu
1041: @node Firstn,,, Primitive function
1042: @subsection @code{Firstn}
1043: @findex Firstn
1044: @table @t
1045: @item Firstn(@var{m},@var{n})
1046: :: Return the first @var{n} elements of @var{m}.
1047: @end table
1048:
1049: */
1050:
1051: /*&ja
1052: @table @var
1053: @item return
1054: $B9TNs$^$?$O%Y%/%H%k(B
1055: @item m
1056: $B9TNs$^$?$O%Y%/%H%k(B
1057: @item n
1058: $B?t(B
1059: @end table
1060:
1061: @itemize @bullet
1062: @item m $B$N:G=i$N(B n $B8D(B. $B$H$/$K(B m $B$,9TNs$N$H$-$O(B, $B3F9T$h$j:G=i$N(B n $B8D$r$H$j$@$7$?(B
1063: $B$b$N$G:n$l$i$?9TNs$rLa$9(B.
1064: @end itemize
1065:
1066: */
1067: /*&en
1068: @table @var
1069: @item return
1070: Matrix or vector
1071: @item m
1072: Matrix or vector
1073: @item n
1074: Number
1075: @end table
1076:
1077: @itemize @bullet
1078: @item The first n elements of m. When m is a matrix, it returns the matrix
1079: consisting of first n elements of rows of m.
1080: @end itemize
1081:
1082: */
1083:
1084: /*&C
1085: @example
1086: In(16)= mm = [[1,2,3],[4,5,6]];
1087: In(17)= Firstn(mm,2):
1088: [[1,2],
1089: [4,5]]
1090: @end example
1091: */
1092:
1093:
1094: /*&ja
1095:
1096: @table @t
1097: @item $B;2>H(B
1.3 takayama 1098: @item Files
1099: @code{lib/restriction/complex.k}
1.1 takayama 1100: @end table
1101: @c %%%%%%%%%%%%%%%%%%%% end of Firstn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1102: */
1103: /*&en
1104:
1105: @table @t
1.3 takayama 1106: @item See also
1107: @item Files
1108: @code{lib/restriction/complex.k}
1109:
1.1 takayama 1110:
1111: @end table
1112: @c %%%%%%%%%%%%%%%%%%%% end of Firstn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1.4 takayama 1113: */
1114:
1115: /*&ja
1116: @c %%%%%%%%%%%%%%%%%%%% start of GKZ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1117: @menu
1118: * GKZ::
1119: @end menu
1120: @node GKZ,,, $B4pK\(B $BH!?t(B
1121: @subsection @code{GKZ}
1122: @findex GKZ
1123: @table @t
1124: @item GKZ(@var{a},@var{b})
1125: :: $B9TNs(B @var{a} $B$H(B $B%Y%/%H%k(B @var{b} $B$GDj$^$k(B, GKZ hypergeometric system $B$rLa$9(B.
1126: @end table
1127:
1128: @table @var
1129: @item return
1130: D $B$N%$%G%"%k$N@8@.85(B
1131: @item a
1132: $B@0?t9TNs(B
1133: @item b
1134: $B@0?t%Y%/%H%k(B
1135: @end table
1136:
1137: @itemize @bullet
1138: @item $B9TNs(B @var{a} $B$H(B $B%Y%/%H%k(B @var{b} $B$GDj$^$k(B, GKZ hypergeometric system $B$rLa$9(B.
1139: @item Gelfand, Kapranov, Zelevinski $B$ND64v2?HyJ,J}Dx<0$rLa$9(B.
1140: @end itemize
1141:
1142: @example
1143: In(3)= GKZ([[1,1,1,1],[0,1,2,3]],[-1,-2]):
1144: [ x1*Dx1+x2*Dx2+x3*Dx3+x4*Dx4+1 , x2*Dx2+2*x3*Dx3+3*x4*Dx4+2 , Dx2^2-Dx1*Dx3 , -Dx3^2+Dx2*Dx4 , Dx2*Dx3-Dx1*Dx4 ]
1145: @end example
1146:
1147: @table @t
1148: @item $B;2>H(B
1149: @item Files
1150: @code{lib/restriction/demo2.k}
1151: @end table
1152: @c %%%%%%%%%%%%%%%%%%%% end of GKZ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1153: */
1154:
1155: /*&ja
1156: @c %%%%%%%%%%%%%%%%%%%% start of Slope %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1157: @menu
1158: * Slope::
1159: @end menu
1160: @node Slope,,, $B4pK\(B $BH!?t(B
1161: @subsection @code{Slope}
1162: @findex Slope
1163: @table @t
1164: @item Slope(@var{i},@var{v},@var{f},@var{v})
1165: :: $BHyJ,J}Dx<07O(B @var{i} $B$N(B slope $B$N=89g$r$b$H$a$k(B.
1166: @end table
1167:
1168: @table @var
1169: @item return
1170: Slope $B$N=89g$H$=$l$rM?$($k(B, weight vectors.
1171: @item i
1172: D $B$N%$%G%"%k$N@8@.85(B
1173: @item v
1174: $B6u4VJQ?t(B
1175: @item f
1176: F-filtration $B$rM?$($k(B weight vector (0,...,0,1,...,1).
1177: @item v
1178: V-filtration $B$rM?$($k(B weight vector.
1179: $B$?$H$($P(B (0,...,0,-1,0,...,0,1).
1180: @end table
1181:
1182: @itemize @bullet
1183: @item @var{i} $B$N(B x_i = 0 $B$K$=$C$?(B, $B86E@$G$N(B slope $B$NA4BN$r$b$H$a$k(B.
1184: @item $B7W;;$K$O(B Assi, Castro, Granger $B$N%"%k%4%j%:%`$rMQ$$$k(B.
1185: @item Geometric slope $B$r7W;;$9$k$?$a(B, radical $B$N7W;;$r$*$3$J$C$F$$$k(B.
1186: $B$3$l$K$O(B ox_asir $B$rMQ$$$F$$$k(B.
1187: @end itemize
1188:
1189: @example
1190: In(13)= a=GKZ([[1,3,7]],[-30]);
1191: In(14)= a:
1192: [ x1*Dx1+3*x2*Dx2+7*x3*Dx3+30 , -Dx1^3+Dx2 , -Dx1*Dx2^2+Dx3 , -Dx2^3+Dx1^2*Dx3 ]
1193: In(15)= Slope(a,[x1,x2,x3],[0,0,0,1,1,1],[0,0,-1,0,0,1]):
1194: [ [ (3)/(4) , [ 0 , 0 , -4 , 3 , 3 , 7 ] ] ]
1195:
1196:
1197: @end example
1198:
1199: @table @t
1200: @item $B;2>H(B
1201: @item Files
1202: @code{lib/restriction/demo2.k}
1203: @end table
1204: @c %%%%%%%%%%%%%%%%%%%% end of Slope %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1.1 takayama 1205: */
1206:
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