Annotation of OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw, Revision 1.7
1.7 ! takayama 1: /*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v 1.6 2019/08/31 06:36:28 takayama Exp $ */
1.1 takayama 2:
3: /*&C
4: @c DO NOT EDIT THIS FILE
5: */
6: /*&C
7: @node SM1 Functions,,, Top
8:
9: */
10: /*&ja
11: @chapter SM1 $BH!?t(B
12:
13: $B$3$N@a$G$O(B sm1 $B$N(B ox $B%5!<%P(B @code{ox_sm1_forAsir}
14: $B$H$N%$%s%?%U%'!<%94X?t$r2r@b$9$k(B.
15: $B$3$l$i$N4X?t$O%U%!%$%k(B @file{sm1.rr} $B$GDj5A$5$l$F$$$k(B.
16: @file{sm1.rr} $B$O(B @file{$(OpenXM_HOME)/lib/asir-contrib} $B$K$"$k(B.
17: $B%7%9%F%`(B @code{sm1} $B$OHyJ,:nMQAG4D$G7W;;$9$k$?$a$N%7%9%F%`$G$"$k(B.
18: $B7W;;Be?t4v2?$N$$$m$$$m$JITJQNL$N7W;;$,HyJ,:nMQAG$N7W;;$K5"Ce$9$k(B.
19: @code{sm1} $B$K$D$$$F$NJ8=q$O(B @code{OpenXM/doc/kan96xx} $B$K$"$k(B.
20:
1.3 takayama 21: $B$J$*(B, sm1 server windows $BHG$O%P%$%J%jG[I[$7$F$$$J$$(B.
22: cygwin $B4D6-$G%=!<%9%3!<%I$+$i%3%s%Q%$%k$7(B, OpenXM/misc/packages/Windows
23: $B$K=>$$JQ99$r2C$($k$H(B sm1 $B%5!<%P$O(Bwindows $B$G$bF0:n$9$k(B.
24:
1.1 takayama 25: $B$H$3$KCG$j$,$J$$$+$.$j$3$N@a$N$9$Y$F$N4X?t$O(B,
26: $BM-M}?t78?t$N<0$rF~NO$H$7$F$&$1$D$1$J$$(B.
27: $B$9$Y$F$NB?9`<0$N78?t$O@0?t$G$J$$$H$$$1$J$$(B.
28:
29: @tex
30: $B6u4V(B
31: $X:={\bf C} \setminus \{ 0, 1 \} = {\bf C} \setminus V(x(x-1))$
32: $B$N%I%i!<%`%3%[%b%m%872C#$N<!85$r7W;;$7$F$_$h$&(B.
33: $X$ $B$OJ?LL$KFs$D$N7j$r$"$1$?6u4V$G$"$k$N$G(B, $BE@(B $x=0$, $x=1$ $B$N$^$o$j$r(B
34: $B$^$o$kFs$D$N%k!<%W$,(B1$B<!85$N%[%b%m%8!<72$N6u4V$r$O$k(B.
35: $B$7$?$,$C$F(B, 1$B<!85%I%i!<%`%3%[%b%m%872$N<!85$O(B $2$ $B$G$"$k(B.
36: @code{sm1} $B$O(B $0$ $B<!85$N%3%[%b%m%872$N<!85$*$h$S(B $1$ $B<!85$N%3%[%b%m%872$N(B
37: $B<!85$rEz$($k(B.
38: @end tex
39: */
40: /*&en
41: @chapter SM1 Functions
42:
43: This chapter describes interface functions for
44: sm1 ox server @code{ox_sm1_forAsir}.
45: These interface functions are defined in the file @file{sm1.rr}.
46: The file @file{sm1.rr} is @*
47: at @file{$(OpenXM_HOME)/lib/asir/contrib-packages}.
48: The system @code{sm1} is a system to compute in the ring of differential
49: operators.
50: Many constructions of invariants
51: in the computational algebraic geometry reduce
52: to constructions in the ring of differential operators.
53: Documents on @code{sm1} are in
54: the directory @code{OpenXM/doc/kan96xx}.
55:
1.3 takayama 56: The sm1 server for windows is not distributed in the binary form.
57: If you need to run it, compile it under the cygwin environment
58: following the Makefile in OpenXM/misc/packages/Windows.
59:
1.1 takayama 60: All the coefficients of input polynomials should be
61: integers for most functions in this section.
62: Other functions accept rational numbers as inputs
63: and it will be explicitely noted in each explanation
64: of these functions.
65:
66:
67:
68: @tex
69: Let us evaluate the dimensions of the de Rham cohomology groups
70: of
71: $X:={\bf C} \setminus \{ 0, 1 \} = {\bf C} \setminus V(x(x-1))$.
72: The space $X$ is a two punctured plane, so two loops that encircles the
73: points $x=0$ and $x=1$ respectively spans the first homology group.
74: Hence, the dimension of the first de Rham cohomology group is $2$.
75: @code{sm1} answers the dimensions of the 0th and the first
76: cohomology groups.
77: @end tex
78: */
79: /*&C
80: @example
81:
82: [283] sm1.deRham([x*(x-1),[x]]);
83: [1,2]
84: @end example
85: */
86: /*&C
87: @noindent
88: The author of @code{sm1} : Nobuki Takayama, @code{takayama@@math.sci.kobe-u.ac.jp} @*
89: The author of sm1 packages : Toshinori Oaku, @code{oaku@@twcu.ac.jp} @*
90: Reference: [SST] Saito, M., Sturmfels, B., Takayama, N.,
91: Grobner Deformations of Hypergeometric Differential Equations,
92: 1999, Springer.
93: @url{http://www.math.kobe-u.ac.jp/KAN}
94: */
95:
96: /*&C
97: @menu
98: * ox_sm1_forAsir::
99: * sm1.start::
100: * sm1.sm1::
101: * sm1.push_int0::
102: * sm1.gb::
103: * sm1.deRham::
104: * sm1.hilbert::
105: * sm1.genericAnn::
106: * sm1.wTensor0::
107: * sm1.reduction::
108: * sm1.xml_tree_to_prefix_string::
109: * sm1.syz::
110: * sm1.mul::
111: * sm1.distraction::
112: * sm1.gkz::
1.2 takayama 113: * sm1.mgkz::
1.1 takayama 114: * sm1.appell1::
115: * sm1.appell4::
116: * sm1.rank::
117: * sm1.auto_reduce::
118: * sm1.slope::
119: * sm1.ahg::
120: * sm1.bfunction::
121: * sm1.generalized_bfunction::
122: * sm1.restriction::
123: * sm1.saturation::
124: @end menu
125: */
126:
127: /*&ja
128: @section @code{ox_sm1_forAsir} $B%5!<%P(B
129: */
130: /*&en
131: @section @code{ox_sm1_forAsir} Server
132: */
133:
134: /*&en
135: @node ox_sm1_forAsir,,, SM1 Functions
136: @subsection @code{ox_sm1_forAsir}
137: @findex ox_sm1_forAsir
138: @table @t
139: @item ox_sm1_forAsir
140: :: @code{sm1} server for @code{asir}.
141: @end table
142: @itemize @bullet
143: @item
144: @code{ox_sm1_forAsir} is the @code{sm1} server started from asir
145: by the command @code{sm1.start}.
146: In the standard setting, @*
147: @code{ox_sm1_forAsir} =
148: @file{$(OpenXM_HOME)/lib/sm1/bin/ox_sm1}
149: +
150: @file{$(OpenXM_HOME)/lib/sm1/callsm1.sm1} (macro file) @*
151: +
152: @file{$(OpenXM_HOME)/lib/sm1/callsm1b.sm1} (macro file) @*
153: The macro files @file{callsm1.sm1} and @file{callsm1b.sm1}
154: are searched from
155: current directory, @code{$(LOAD_SM1_PATH)},
156: @code{$(OpenXM_HOME)/lib/sm1},
157: @code{/usr/local/lib/sm1}
158: in this order.
159: @item Note for programmers: See the files
160: @file{$(OpenXM_HOME)/src/kxx/oxserver00.c},
161: @file{$(OpenXM_HOME)/src/kxx/sm1stackmachine.c}
162: to build your own server by reading @code{sm1} macros.
163: @end itemize
164: */
165: /*&ja
166: @node ox_sm1_forAsir,,, SM1 Functions
167: @subsection @code{ox_sm1_forAsir}
168: @findex ox_sm1_forAsir
169: @table @t
170: @item ox_sm1_forAsir
171: :: @code{asir} $B$N$?$a$N(B @code{sm1} $B%5!<%P(B.
172: @end table
173: @itemize @bullet
174: @item
175: $B%5!<%P(B @code{ox_sm1_forAsir} $B$O(B @code{asir} $B$h$j%3%^%s%I(B
176: @code{sm1.start} $B$G5/F0$5$l$k(B @code{sm1} $B%5!<%P$G$"$k(B.
177:
178: $BI8=`E*@_Dj$G$O(B, @*
179: @code{ox_sm1_forAsir} =
180: @file{$(OpenXM_HOME)/lib/sm1/bin/ox_sm1}
181: +
182: @file{$(OpenXM_HOME)/lib/sm1/callsm1.sm1} (macro file) @*
183: +
184: @file{$(OpenXM_HOME)/lib/sm1/callsm1b.sm1} (macro file) @*
185: $B$G$"$j(B, $B$3$l$i$N%^%/%m%U%!%$%k$O(B, $B0lHL$K$O(B
186: current directory, @code{$(LOAD_SM1_PATH)},
187: @code{$(OpenXM_HOME)/lib/sm1},
188: @code{/usr/local/lib/sm1}
189: $B$N=gHV$G$5$,$5$l$k(B.
190: @item $B%W%m%0%i%^!<$N$?$a$N%N!<%H(B:
191: @code{sm1} $B%^%/%m$rFI$_9~$s$G<+J,FH<+$N%5!<%P$r:n$k$K$O(B
192: $B<!$N%U%!%$%k$b8+$h(B
193: @file{$(OpenXM_HOME)/src/kxx/oxserver00.c},
194: @file{$(OpenXM_HOME)/src/kxx/sm1stackmachine.c}
195: @end itemize
196: */
197:
198:
199: /*&ja
200: @section $BH!?t0lMw(B
201: */
202: /*&en
203: @section Functions
204: */
205:
206: /*&en
207: @c sort-sm1.start
208: @node sm1.start,,, SM1 Functions
209: @subsection @code{sm1.start}
210: @findex sm1.start
211: @table @t
212: @item sm1.start()
213: :: Start @code{ox_sm1_forAsir} on the localhost.
214: @end table
215:
216: @table @var
217: @item return
218: Integer
219: @end table
220:
221: @itemize @bullet
222: @item Start @code{ox_sm1_forAsir} on the localhost.
223: It returns the descriptor of @code{ox_sm1_forAsir}.
224: @item Set @code{Xm_noX = 1} to start @code{ox_sm1_forAsir}
225: without a debug window.
226: @item You might have to set suitable orders of variable by the command
227: @code{ord}. For example,
228: when you are working in the
229: ring of differential operators on the variable @code{x} and @code{dx}
230: (@code{dx} stands for
231: @tex $\partial/\partial x$
232: @end tex
233: ),
234: @code{sm1} server assumes that
235: the variable @code{dx} is collected to the right and the variable
236: @code{x} is collected to the left in the printed expression.
237: In the example below, you must not use the variable @code{cc}
238: for computation in @code{sm1}.
239: @item The variables from @code{a} to @code{z} except @code{d} and @code{o}
240: and @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},
241: @code{z0}, ..., @code{z20} can be used as variables for ring of
242: differential operators in default. (cf. @code{Sm1_ord_list} in @code{sm1}).
243: @item The descriptor is stored in @code{static Sm1_proc}.
244: The descriptor can be obtained by the function
245: @code{sm1.get_Sm1_proc()}.
246: @end itemize
247: */
248: /*&ja
249: @c sort-sm1.start
250: @node sm1.start,,, SM1 Functions
251: @subsection @code{sm1.start}
252: @findex sm1.start
253: @table @t
254: @item sm1.start()
255: :: localhost $B$G(B @code{ox_sm1_forAsir} $B$r%9%?!<%H$9$k(B.
256: @end table
257:
258: @table @var
259: @item return
260: $B@0?t(B
261: @end table
262:
263: @itemize @bullet
264: @item localhost $B$G(B @code{ox_sm1_forAsir} $B$r%9%?!<%H$9$k(B.
265: $B%5!<%P(B @code{ox_sm1_forAsir} $B$N<1JLHV9f$rLa$9(B.
266: @item @code{Xm_noX = 1} $B$H$*$/$H%5!<%P(B @code{ox_sm1_forAsir} $B$r%G%P%C%0MQ$N(B
267: $B%&%#%s%I%&$J$7$K5/F0$G$-$k(B.
268: @item $B%3%^%s%I(B @code{ord} $B$rMQ$$$FJQ?t=g=x$r@5$7$/@_Dj$7$F$*$/I,MW$,(B
269: $B$"$k(B.
270: $B$?$H$($P(B,
271: $BJQ?t(B @code{x} $B$H(B @code{dx} $B>e$NHyJ,:nMQAG4D(B
272: (@code{dx} $B$O(B
273: @tex $\partial/\partial x$
274: @end tex
275: $B$KBP1~(B)
276: $B$G7W;;$7$F$$$k$H$-(B,
277: @code{sm1} $B%5!<%P$O<0$r0u:~$7$?$H$-(B,
278: $BJQ?t(B @code{dx} $B$O1&B&$K=8$a$lJQ?t(B
279: @code{x} $B$O:8B&$K$"$D$a$i$l$F$$$k$H2>Dj$7$F$$$k(B.
280: $B<!$NNc$G$O(B, $BJQ?t(B @code{cc} $B$r(B @code{sm1} $B$G$N7W;;$N$?$a$KMQ$$$F$O$$$1$J$$(B.
281: @item @code{a} $B$h$j(B @code{z} $B$N$J$+$G(B, @code{d} $B$H(B @code{o} $B$r=|$$$?$b$N(B,
282: $B$=$l$+$i(B, @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},
283: @code{z0}, ..., @code{z20} $B$O(B, $B%G%U%)!<%k%H$GHyJ,:nMQAG4D$NJQ?t$H$7$F(B
284: $B;H$($k(B (cf. @code{Sm1_ord_list} in @code{sm1}).
285: @item $B<1JLHV9f$O(B @code{static Sm1_proc} $B$K3JG<$5$l$k(B.
286: $B$3$N<1JLHV9f$O4X?t(B @code{sm1.get_Sm1_proc()} $B$G$H$j$@$9$3$H$,$G$-$k(B.
287: @end itemize
288: */
289: /*&C
290: @example
291: [260] ord([da,a,db,b]);
292: [da,a,db,b,dx,dy,dz,x,y,z,dt,ds,t,s,u,v,w,
293: ......... omit ..................
294: ]
295: [261] a*da;
296: a*da
297: [262] cc*dcc;
298: dcc*cc
299: [263] sm1.mul(da,a,[a]);
300: a*da+1
301: [264] sm1.mul(a,da,[a]);
302: a*da
303: @end example
304: */
305: /*&en
306: @table @t
307: @item Reference
308: @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
309: @code{ord}
310: @end table
311: */
312: /*&ja
313: @table @t
314: @item $B;2>H(B
315: @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
316: @code{ord}
317: @end table
318: */
319:
320:
321:
322: /*&en
323: @c sort-sm1
324: @node sm1.sm1,,, SM1 Functions
325: @subsection @code{sm1.sm1}
326: @findex sm1.sm1
327: @table @t
328: @item sm1.sm1(@var{p},@var{s})
329: :: ask the @code{sm1} server to execute the command string @var{s}.
330: @end table
331:
332: @table @var
333: @item return
334: Void
335: @item p
336: Number
337: @item s
338: String
339: @end table
340:
341: @itemize @bullet
342: @item It asks the @code{sm1} server of the descriptor number @var{p}
343: to execute the command string @var{s}.
344: (In the next example, the descriptor number is 0.)
345: @end itemize
346: */
347: /*&ja
348: @node sm1.sm1,,, SM1 Functions
349: @subsection @code{sm1.sm1}
350: @findex sm1.sm1
351: @table @t
352: @item sm1.sm1(@var{p},@var{s})
353: :: $B%5!<%P(B @code{sm1} $B$K%3%^%s%INs(B @var{s} $B$r<B9T$7$F$/$l$k$h$&$K$?$N$`(B.
354: @end table
355:
356: @table @var
357: @item return
358: $B$J$7(B
359: @item p
360: $B?t(B
361: @item s
362: $BJ8;zNs(B
363: @end table
364:
365: @itemize @bullet
366: @item $B<1JLHV9f(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B
367: $B%3%^%s%INs(B @var{s} $B$r<B9T$7$F$/$l$k$h$&$KMj$`(B.
368: ($B<!$NNc$G$O(B, $B<1JLHV9f(B 0)
369: @end itemize
370: */
371: /*&C
372: @example
373: [261] sm1.sm1(0," ( (x-1)^2 ) . ");
374: 0
375: [262] ox_pop_string(0);
376: x^2-2*x+1
377: [263] sm1.sm1(0," [(x*(x-1)) [(x)]] deRham ");
378: 0
379: [264] ox_pop_string(0);
380: [1 , 2]
381: @end example
382: */
383:
384: /*&ja
385: @table @t
386: @item $B;2>H(B
387: @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
388: @end table
389: */
390: /*&en
391: @table @t
392: @item Reference
393: @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
394: @end table
395: */
396:
397:
398: /*&en
399: @c sort-sm1.push_int0
400: @node sm1.push_int0,,, SM1 Functions
401: @subsection @code{sm1.push_int0}
402: @findex sm1.push_int0
403: @table @t
404: @item sm1.push_int0(@var{p},@var{f})
405: :: push the object @var{f} to the server with the descriptor number @var{p}.
406: @end table
407:
408: @table @var
409: @item return
410: Void
411: @item p
412: Number
413: @item f
414: Object
415: @end table
416:
417: @itemize @bullet
418: @item When @code{type(@var{f})} is 2 (recursive polynomial),
419: @var{f} is converted to a string (type == 7)
420: and is sent to the server by @code{ox_push_cmo}.
421: @item When @code{type(@var{f})} is 0 (zero),
422: it is translated to the 32 bit integer zero
423: on the server.
424: Note that @code{ox_push_cmo(@var{p},0)} sends @code{CMO_NULL} to the server.
425: In other words, the server does not get the 32 bit integer 0 nor
426: the bignum 0.
427: @item @code{sm1} integers are classfied into the 32 bit integer and
428: the bignum.
429: When @code{type(@var{f})} is 1 (number), it is translated to the
430: 32 bit integer on the server.
431: Note that @code{ox_push_cmo(@var{p},1234)} send the bignum 1234 to the
432: @code{sm1} server.
433: @item In other cases, @code{ox_push_cmo} is called without data conversion.
434: @end itemize
435: */
436: /*&ja
437: @c sort-sm1.push_int0
438: @node sm1.push_int0,,, SM1 Functions
439: @subsection @code{sm1.push_int0}
440: @findex sm1.push_int0
441: @table @t
442: @item sm1.push_int0(@var{p},@var{f})
443: :: $B%*%V%8%'%/%H(B @var{f} $B$r<1JL;R(B @var{p} $B$N%5!<%P$XAw$k(B.
444: @end table
445:
446: @table @var
447: @item return
448: $B$J$7(B
449: @item p
450: $B?t(B
451: @item f
452: $B%*%V%8%'%/%H(B
453: @end table
454:
455: @itemize @bullet
456: @item @code{type(@var{f})} $B$,(B 2 ($B:F5"B?9`<0(B) $B$N$H$-(B,
457: @var{f} $B$OJ8;zNs(B (type == 7) $B$KJQ49$5$l$F(B,
458: @code{ox_push_cmo} $B$rMQ$$$F%5!<%P$XAw$i$l$k(B.
459: @item @code{type(@var{f})} $B$,(B 0 (zero) $B$N$H$-$O(B,
460: $B%5!<%P>e$G$O(B, 32 bit $B@0?t$H2r<a$5$l$k(B.
461: $B$J$*(B @code{ox_push_cmo(P,0)} $B$O%5!<%P$KBP$7$F(B @code{CMO_NULL}
462: $B$r$*$/$k$N$G(B, $B%5!<%PB&$G$O(B, 32 bit $B@0?t$r<u$1<h$k$o$1$G$O$J$$(B.
463: @item @code{sm1} $B$N@0?t$O(B, 32 bit $B@0?t$H(B bignum $B$K$o$1$k$3$H$,$G$-$k(B.
464: @code{type(@var{f})} $B$,(B 1 ($B?t(B)$B$N$H$-(B, $B$3$N4X?t$O(B 32 bit integer $B$r%5!<%P$K(B
465: $B$*$/$k(B.
466: @code{ox_push_cmo(@var{p},1234)} $B$O(B bignum $B$N(B 1234 $B$r(B
467: @code{sm1} $B%5!<%P$K$*$/$k$3$H$KCm0U$7$h$&(B.
468: @item $B$=$NB>$N>l9g$K$O(B @code{ox_push_cmo} $B$r%G!<%?7?$NJQ49$J$7$K8F$S=P$9(B.
469: @end itemize
470: */
471: /*&C
472: @example
473: [219] P=sm1.start();
474: 0
475: [220] sm1.push_int0(P,x*dx+1);
476: 0
477: [221] A=ox_pop_cmo(P);
478: x*dx+1
479: [223] type(A);
480: 7 (string)
481: @end example
482:
483: @example
484: [271] sm1.push_int0(0,[x*(x-1),[x]]);
485: 0
486: [272] ox_execute_string(0," deRham ");
487: 0
488: [273] ox_pop_cmo(0);
489: [1,2]
490: @end example
491: */
492: /*&en
493: @table @t
494: @item Reference
495: @code{ox_push_cmo}
496: @end table
497: */
498: /*&ja
499: @table @t
500: @item Reference
501: @code{ox_push_cmo}
502: @end table
503: */
504:
505:
506:
507: /*&en
508: @c sort-sm1.gb
509: @node sm1.gb,,, SM1 Functions
510: @subsection @code{sm1.gb}
511: @findex sm1.gb
512: @findex sm1.gb_d
513: @table @t
1.6 takayama 514: @item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r},needBack=@var{n})
1.1 takayama 515: :: computes the Grobner basis of @var{f} in the ring of differential
516: operators with the variable @var{v}.
517: @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
518: :: computes the Grobner basis of @var{f} in the ring of differential
519: operators with the variable @var{v}.
520: The result will be returned as a list of distributed polynomials.
521: @end table
522:
523: @table @var
524: @item return
525: List
526: @item p, q, r
527: Number
528: @item f, v, w
529: List
530: @end table
531:
532: @itemize @bullet
533: @item
534: It returns the Grobner basis of the set of polynomials @var{f}
535: in the ring of deferential operators with the variables @var{v}.
536: @item
537: The weight vectors are given by @var{w}, which can be omitted.
538: If @var{w} is not given,
539: the graded reverse lexicographic order will be used to compute Grobner basis.
540: @item
541: The return value of @code{sm1.gb}
542: is the list of the Grobner basis of @var{f} and the initial
543: terms (when @var{w} is not given) or initial ideal (when @var{w} is given).
544: @item
545: @code{sm1.gb_d} returns the results by a list of distributed polynomials.
546: Monomials in each distributed polynomial are ordered in the given order.
547: The return value consists of
548: [variable names, order matrix, grobner basis in districuted polynomials,
549: initial monomials or initial polynomials].
550: @item
551: When a non-term order is given, the Grobner basis is computed in
552: the homogenized Weyl algebra (See Section 1.2 of the book of SST).
553: The homogenization variable h is automatically added.
554: @item
555: When the optional variable @var{q} is set, @code{sm1.gb} returns,
556: as the third return value, a list of
557: the Grobner basis and the initial ideal
558: with sums of monomials sorted by the given order.
559: Each polynomial is expressed as a string temporally for now.
560: When the optional variable @var{r} is set to one,
561: the polynomials are dehomogenized (,i.e., h is set to 1).
1.5 takayama 562: @item If you want to have a reduced basis or compute the initial form ideal exactly,
563: execute sm1.auto_reduce(1) before executing this function.
1.7 ! takayama 564: @item When the needBack option @var{n} is 1, it returns the answer is a different format as [groebner basis,initial, gb,1,all,[groebner basis, backward transformation]]
1.1 takayama 565: @end itemize
566: */
567: /*&ja
568: @c sort-sm1.gb
569: @node sm1.gb,,, SM1 Functions
570: @subsection @code{sm1.gb}
571: @findex sm1.gb
572: @findex sm1.gb_d
573: @table @t
1.6 takayama 574: @item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r},needBack=@var{n})
1.1 takayama 575: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
576: @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
577: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. $B7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.
578: @end table
579:
580: @table @var
581: @item return
582: $B%j%9%H(B
583: @item p, q, r
584: $B?t(B
585: @item f, v, w
586: $B%j%9%H(B
587: @end table
588:
589: @itemize @bullet
590: @item
591: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
592: @item
593: Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
594: $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
595: $B%V%l%V%J4pDl$r7W;;$9$k(B.
596: @item
597: @code{sm1.gb} $B$NLa$jCM$O(B @var{f} $B$N%0%l%V%J4pDl$*$h$S%$%K%7%c%k%b%N%_%"%k(B
598: ( @var{w} $B$,$J$$$H$-(B ) $B$^$?$O(B $B%$%K%7%!%kB?9`<0(B ( @var{w} $B$,M?$($i$?$H$-(B)
599: $B$N%j%9%H$G$"$k(B.
600: @item
601: @code{sm1.gb_d} $B$O7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.
602: $BB?9`<0$NCf$K8=$l$k%b%N%_%"%k$O%0%l%V%J4pDl$r7W;;$9$k$H$-$KM?$($i$?=g=x$G%=!<%H$5$l$F$$$k(B.
603: $BLa$jCM$O(B
604: [$BJQ?tL>$N%j%9%H(B, $B=g=x$r$-$a$k9TNs(B, $B%0%l%V%J4pDl(B, $B%$%K%7%c%k%b%N%_%"%k$^$?$O%$%K%7%!%kB?9`<0(B]
605: $B$G$"$k(B.
606: @item
607: Term order $B$G$J$$=g=x$,M?$($i$l$?>l9g$O(B, $BF1<!2=%o%$%kBe?t$G%0%l%V%J4pDl$,7W;;$5$l$k(B (SST $B$NK\$N(B Section 1.2 $B$r8+$h(B).
608: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
609: @item $B%*%W%7%g%J%kJQ?t(B @var{q} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
610: 3 $BHVL\$NLa$jCM$H$7$F(B, $B%0%l%V%J4pDl$*$h$S%$%K%7%!%k$N%j%9%H$,(B
611: $BM?$($i$l$?=g=x$G%=!<%H$5$l$?%b%N%_%"%k$NOB$H$7$FLa$5$l$k(B.
612: $B$$$^$N$H$3$m$3$NB?9`<0$O(B, $BJ8;zNs$GI=8=$5$l$k(B.
613: $B%*%W%7%g%J%kJQ?t(B @var{r} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
614: $BLa$jB?9`<0$O(B dehomogenize $B$5$l$k(B ($B$9$J$o$A(B h $B$K(B 1 $B$,BeF~$5$l$k(B).
1.5 takayama 615: @item Reduced $B%0%l%V%J!<4pDl$^$?$O(B in_w $B$r7W;;$7$?$$$H$-$O(B, $B$3$N4X?t$N<B9T$NA0$K(B
616: sm1.auto_reduce(1) $B$r<B9T$7$F$*$/$3$H(B.
1.6 takayama 617: @item needBack $B%*%W%7%g%s$,(B 1 $B$N;~$O(B, $BB>$N>l9g$H$O0[$J$k7A<0(B
1.7 ! takayama 618: [groebner basis, initial, gb,1,all, [groebner basis, backward transformation]]
1.6 takayama 619: $B$GEz$($rLa$9(B. (sm1 $B$N(B getAttribute $B$r;2>H(B)
1.1 takayama 620: @end itemize
621: */
622: /*&C
623: @example
624: [293] sm1.gb([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
625: [[x*dx+y*dy-1,y^2*dy^2+2],[x*dx,y^2*dy^2]]
626: @end example
627: */
628: /*&en
629: In the example above,
630: @tex the set $\{ x \partial_x + y \partial_y -1,
631: y^2 \partial_y^2+2\}$
632: is the Gr\"obner basis of the input with respect to the
633: graded reverse lexicographic order such that
634: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$.
635: The set $\{x \partial_x, y^2 \partial_y\}$ is the leading monomials
636: (the initial monominals) of the Gr\"obner basis.
637: @end tex
638: */
639: /*&ja
640: $B>e$NNc$K$*$$$F(B,
641: @tex $B=89g(B $\{ x \partial_x + y \partial_y -1,
642: y^2 \partial_y^2+2\}$
643: $B$O(B
644: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$
645: $B$G$"$k$h$&$J(B
646: graded reverse lexicographic order $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
647: $B=89g(B $\{x \partial_x, y^2 \partial_y\}$ $B$O%0%l%V%J4pDl$N3F85$K(B
648: $BBP$9$k(B leading monomial (initial monomial) $B$G$"$k(B.
649: @end tex
650: */
651: /*&C
652: @example
653: [294] sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]);
654: [[dx+dy^3-4*dy,-dy^4+4*dy^2-1],[dx,-dy^4]]
655: @end example
656: */
657: /*&en
658: In the example above, two monomials
659: @tex
660: $m = x^a y^b \partial_x^c \partial_y^d$ and
661: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
662: are firstly compared by the weight vector
663: {\tt (dx,dy,x,y) = (50,2,1,0)}
664: (i.e., $m$ is larger than $m'$ if $50c+2d+a > 50c'+2d'+a'$)
665: and when the comparison is tie, then these are
666: compared by the reverse lexicographic order
667: (i.e., if $50c+2d+a = 50c'+2d'+a'$, then use the reverse lexicogrpahic order).
668: @end tex
669: */
670: /*&ja
671: $B>e$NNc$K$*$$$FFs$D$N%b%N%_%"%k(B
672: @tex
673: $m = x^a y^b \partial_x^c \partial_y^d$ $B$*$h$S(B
674: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
675: $B$O:G=i$K(B weight vector
676: {\tt (dx,dy,x,y) = (50,2,1,0)} $B$rMQ$$$FHf3S$5$l$k(B
677: ($B$D$^$j(B $m$ $B$O(B $50c+2d+a > 50c'+2d'+a'$ $B$N$H$-(B
678: $m'$ $B$h$jBg$-$$(B )
679: $B<!$K$3$NHf3S$G>!Ii$,$D$+$J$$$H$-$O(B reverse lexicographic order $B$GHf3S$5$l$k(B
680: ($B$D$^$j(B $50c+2d+a = 50c'+2d'+a'$ $B$N$H$-(B reverse lexicographic order $B$GHf3S(B
681: $B$5$l$k(B).
682: @end tex
683: */
684: /*&C
685: @example
686: [294] F=sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);
687: map(print,F[2][0])$
688: map(print,F[2][1])$
689: @end example
690: */
691: /*&C
692: @example
693: [595]
694: sm1.gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],
695: [x,y],[[dx,1,x,-1],[dy,1]]]);
696:
697: [[x*dx^2+(y*dy-h^2)*dx-h^3,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx-h^3*dy],
698: [x*dx^2+(y*dy-h^2)*dx,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx]]
699:
700: [596]
701: sm1.gb_d([["dx (x dx +y dy-2)-1","dy (x dx + y dy -2)-1"],
702: "x,y",[[dx,1,x,-1],[dy,1]]]);
703: [[[e0,x,y,H,E,dx,dy,h],
704: [[0,-1,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],
705: [0,1,1,1,1,1,1,0],[0,0,0,0,0,0,-1,0],[0,0,0,0,0,-1,0,0],
706: [0,0,0,0,-1,0,0,0],[0,0,0,-1,0,0,0,0],[0,0,-1,0,0,0,0,0],
707: [0,0,0,0,0,0,0,1]]],
708: [[(1)*<<0,0,1,0,0,1,1,0>>+(1)*<<0,1,0,0,0,2,0,0>>+(-1)*<<0,0,0,0,0,1,0,2>>+(-1)*
709: <<0,0,0,0,0,0,0,3>>,(1)*<<0,0,1,0,0,0,2,0>>+(1)*<<0,1,0,0,0,1,1,0>>+(-1)*<<0,0,0
710: ,0,0,0,1,2>>+(-1)*<<0,0,0,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>+(-1)*<<0,0,0,0,0,0
711: ,1,3>>],
712: [(1)*<<0,0,1,0,0,1,1,0>>+(1)*<<0,1,0,0,0,2,0,0>>+(-1)*<<0,0,0,0,0,1,0,2>>,(1)*<
713: <0,0,1,0,0,0,2,0>>+(1)*<<0,1,0,0,0,1,1,0>>+(-1)*<<0,0,0,0,0,0,1,2>>+(-1)*<<0,0,0
714: ,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]]
715: @end example
716: */
1.6 takayama 717: /*&C
718: @example
719: [1834] sm1.gb([[dx^2-x,dx],[x]] | needBack=1);
1.7 ! takayama 720: [[dx,dx^2-x,1],[dx,dx^2,1],gb,1,all,[[dx,dx^2-x,1],[[0,1],[1,0],[-dx,dx^2-x]]]]
1.6 takayama 721: @end example
722: */
1.1 takayama 723:
724: /*&en
725: @table @t
726: @item Reference
1.5 takayama 727: @code{sm1.auto_reduce}, @code{sm1.reduction}, @code{sm1.rat_to_p}
1.1 takayama 728: @end table
729: */
730: /*&ja
731: @table @t
732: @item $B;2>H(B
1.5 takayama 733: @code{sm1.auto_reduce}, @code{sm1.reduction}, @code{sm1.rat_to_p}
1.1 takayama 734: @end table
735: */
736:
737:
738:
739: /*&en
740: @c sort-sm1.deRham
741: @node sm1.deRham,,, SM1 Functions
742: @subsection @code{sm1.deRham}
743: @findex sm1.deRham
744: @table @t
745: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
746: :: ask the server to evaluate the dimensions of the de Rham cohomology groups
747: of C^n - (the zero set of @var{f}=0).
748: @end table
749:
750: @table @var
751: @item return
752: List
753: @item p
754: Number
755: @item f
756: String or polynomial
757: @item v
758: List
759: @end table
760:
761: @itemize @bullet
762: @item It returns the dimensions of the de Rham cohomology groups
763: of X = C^n \ V(@var{f}).
764: In other words, it returns
765: [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)].
766: @item @var{v} is a list of variables. n = @code{length(@var{v})}.
767: @item
768: @code{sm1.deRham} requires huge computer resources.
769: For example, @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
770: is already very hard.
771: @item
772: To efficiently analyze the roots of b-function, @code{ox_asir} should be used
773: from @code{ox_sm1_forAsir}.
774: It is recommended to load the communication module for @code{ox_asir}
775: by the command @*
776: @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
777: This command is automatically executed when @code{ox_sm1_forAsir} is started.
778: @item If you make an interruption to the function @code{sm1.deRham}
779: by @code{ox_reset(sm1.get_Sm1_proc());}, the server might get out of the standard
780: mode. So, it is strongly recommended to execute the command
781: @code{ox_shutdown(sm1.get_Sm1_proc());} to interrupt and restart the server.
782: @end itemize
783: */
784: /*&ja
785: @c sort-sm1.deRham
786: @node sm1.deRham,,, SM1 Functions
787: @subsection @code{sm1.deRham}
788: @findex sm1.deRham
789: @table @t
790: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
791: :: $B6u4V(B C^n - (the zero set of @var{f}=0) $B$N%I%i!<%`%3%[%b%m%872$N<!85$r7W;;$7$F$/$l$k$h$&$K%5!<%P$KMj$`(B.
792: @end table
793:
794: @table @var
795: @item return
796: $B%j%9%H(B
797: @item p
798: $B?t(B
799: @item f
800: $BJ8;zNs(B $B$^$?$O(B $BB?9`<0(B
801: @item v
802: $B%j%9%H(B
803: @end table
804:
805: @itemize @bullet
806: @item $B$3$NH!?t$O6u4V(B X = C^n \ V(@var{f}) $B$N%I%i!<%`%3%[%b%m%872$N<!85$r7W;;$9$k(B.
807: $B$9$J$o$A(B,
808: [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)]
809: $B$rLa$9(B.
810: @item @var{v} $B$OJQ?t$N%j%9%H(B. n = @code{length(@var{v})} $B$G$"$k(B.
811: @item
812: @code{sm1.deRham} $B$O7W;;5!$N;q8;$rBgNL$K;HMQ$9$k(B.
813: $B$?$H$($P(B @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
814: $B$N7W;;$9$i$9$G$KHs>o$KBgJQ$G$"$k(B.
815: @item
816: b-$B4X?t$N:,$r8zN($h$/2r@O$9$k$K$O(B, @code{ox_asir} $B$,(B @code{ox_sm1_forAsir}
817: $B$h$j;HMQ$5$l$k$Y$-$G$"$k(B. $B%3%^%s%I(B @*
818: @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
819: $B$rMQ$$$F(B, @code{ox_asir} $B$H$NDL?.%b%8%e!<%k$r$"$i$+$8$a%m!<%I$7$F$*$/$H$h$$(B.
820: $B$3$N%3%^%s%I$O(B @code{ox_asir_forAsir} $B$N%9%?!<%H;~$K<+F0E*$K<B9T$5$l$F$$$k(B.
821: @item
822: @code{sm1.deRham} $B$r(B @code{ox_reset(sm1.get_Sm1_proc());} $B$GCfCG$9$k$H(B,
823: $B0J8e(B sm1 $B%5!<%P$,HsI8=`%b!<%I$KF~$jM=4|$7$J$$F0:n$r$9$k>l9g(B
824: $B$,$"$k$N$G(B, $B%3%^%s%I(B @code{ox_shutdown(sm1.get_Sm1_proc());} $B$G(B, @code{ox_sm1_forAsir}
825: $B$r0l;~(B shutdown $B$7$F%j%9%?!<%H$7$?J}$,0BA4$G$"$k(B.
826: @end itemize
827: */
828: /*&C
829: @example
830: [332] sm1.deRham([x^3-y^2,[x,y]]);
831: [1,1,0]
832: [333] sm1.deRham([x*(x-1),[x]]);
833: [1,2]
834: @end example
835: */
836: /*&en
837: @table @t
838: @item Reference
839: @code{sm1.start}, @code{deRham} (sm1 command)
840: @item Algorithm:
841: Oaku, Takayama, An algorithm for de Rham cohomology groups of the
842: complement of an affine variety via D-module computation,
843: Journal of pure and applied algebra 139 (1999), 201--233.
844: @end table
845: */
846: /*&ja
847: @table @t
848: @item $B;2>H(B
849: @code{sm1.start}, @code{deRham} (sm1 command)
850: @item Algorithm:
851: Oaku, Takayama, An algorithm for de Rham cohomology groups of the
852: complement of an affine variety via D-module computation,
853: Journal of pure and applied algebra 139 (1999), 201--233.
854: @end table
855: */
856:
857:
858:
859:
860: /*&en
861: @c sort-sm1.hilbert
862: @node sm1.hilbert,,, SM1 Functions
863: @subsection @code{sm1.hilbert}
864: @findex sm1.hilbert
865: @findex hilbert_polynomial
866: @table @t
867: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
868: :: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
869: @item hilbert_polynomial(@var{f},@var{v})
870: :: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
871: @end table
872:
873: @table @var
874: @item return
875: Polynomial
876: @item p
877: Number
878: @item f, v
879: List
880: @end table
881:
882: @itemize @bullet
883: @item It returns the Hilbert polynomial h(k) of the set of polynomials
884: @var{f}
885: with respect to the set of variables @var{v}.
886: @item
887: h(k) = dim_Q F_k/I \cap F_k where F_k the set of polynomials of which
888: degree is less than or equal to k and I is the ideal generated by the
889: set of polynomials @var{f}.
890: @item
891: Note for sm1.hilbert:
892: For an efficient computation, it is preferable that
893: the set of polynomials @var{f} is a set of monomials.
894: In fact, this function firstly compute a Grobner basis of @var{f}, and then
895: compute the Hilbert polynomial of the initial monomials of the basis.
896: If the input @var{f} is already a Grobner
897: basis, a Grobner basis is recomputed in this function,
898: which is a waste of time and Grobner basis computation in the ring of
899: polynomials in @code{sm1} is slower than in @code{asir}.
900: @end itemize
901: */
902: /*&ja
903: @c sort-sm1.hilbert
904: @node sm1.hilbert,,, SM1 Functions
905: @subsection @code{sm1.hilbert}
906: @findex sm1.hilbert
907: @findex hilbert_polynomial
908: @table @t
909: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
910: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
911: @item hilbert_polynomial(@var{f},@var{v})
912: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
913: @end table
914:
915: @table @var
916: @item return
917: $BB?9`<0(B
918: @item p
919: $B?t(B
920: @item f, v
921: $B%j%9%H(B
922: @end table
923:
924: @itemize @bullet
925: @item $BB?9`<0$N=89g(B @var{f} $B$NJQ?t(B @var{v} $B$K$+$s$9$k%R%k%Y%k%HB?9`<0(B h(k)
926: $B$r7W;;$9$k(B.
927: @item
928: h(k) = dim_Q F_k/I \cap F_k $B$3$3$G(B F_k $B$O<!?t$,(B k $B0J2<$G$"$k$h$&$J(B
929: $BB?9`<0$N=89g$G$"$k(B. I $B$OB?9`<0$N=89g(B @var{f} $B$G@8@.$5$l$k%$%G%"%k$G$"$k(B.
930: @item
931: sm1.hilbert $B$K$+$s$9$k%N!<%H(B:
932: $B8zN($h$/7W;;$9$k$K$O(B @var{f} $B$O%b%N%_%"%k$N=89g$K$7$?J}$,$$$$(B.
933: $B<B:](B, $B$3$NH!?t$O$^$:(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$7(B, $B$=$l$+$i$=$N(B initial
934: monomial $BC#$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
935: $B$7$?$,$C$F(B, $BF~NO(B @var{f} $B$,$9$G$K%0%l%V%J4pDl$@$H$3$NH!?t$N$J$+$G$b$&0lEY(B
936: $B%0%l%V%J4pDl$N7W;;$,$*$3$J$o$l$k(B. $B$3$l$O;~4V$NL5BL$G$"$k$7(B, @code{sm1} $B$N(B
937: $BB?9`<0%0%l%V%J4pDl7W;;$O(B @code{asir} $B$h$jCY$$(B.
938: @end itemize
939: */
940:
941: /*&C
942: @example
943:
944: [346] load("katsura")$
945: [351] A=hilbert_polynomial(katsura(5),[u0,u1,u2,u3,u4,u5]);
946: 32
947:
948: @end example
949:
950: @example
951: [279] load("katsura")$
952: [280] A=gr(katsura(5),[u0,u1,u2,u3,u4,u5],0)$
953: [281] dp_ord();
954: 0
955: [282] B=map(dp_ht,map(dp_ptod,A,[u0,u1,u2,u3,u4,u5]));
956: [(1)*<<1,0,0,0,0,0>>,(1)*<<0,0,0,2,0,0>>,(1)*<<0,0,1,1,0,0>>,(1)*<<0,0,2,0,0,0>>,
957: (1)*<<0,1,1,0,0,0>>,(1)*<<0,2,0,0,0,0>>,(1)*<<0,0,0,1,1,1>>,(1)*<<0,0,0,1,2,0>>,
958: (1)*<<0,0,1,0,2,0>>,(1)*<<0,1,0,0,2,0>>,(1)*<<0,1,0,1,1,0>>,(1)*<<0,0,0,0,2,2>>,
959: (1)*<<0,0,1,0,1,2>>,(1)*<<0,1,0,0,1,2>>,(1)*<<0,1,0,1,0,2>>,(1)*<<0,0,0,0,3,1>>,
960: (1)*<<0,0,0,0,4,0>>,(1)*<<0,0,0,0,1,4>>,(1)*<<0,0,0,1,0,4>>,(1)*<<0,0,1,0,0,4>>,
961: (1)*<<0,1,0,0,0,4>>,(1)*<<0,0,0,0,0,6>>]
962: [283] C=map(dp_dtop,B,[u0,u1,u2,u3,u4,u5]);
963: [u0,u3^2,u3*u2,u2^2,u2*u1,u1^2,u5*u4*u3,u4^2*u3,u4^2*u2,u4^2*u1,u4*u3*u1,
964: u5^2*u4^2,u5^2*u4*u2,u5^2*u4*u1,u5^2*u3*u1,u5*u4^3,u4^4,u5^4*u4,u5^4*u3,
965: u5^4*u2,u5^4*u1,u5^6]
966: [284] sm1.hilbert([C,[u0,u1,u2,u3,u4,u5]]);
967: 32
968: @end example
969: */
970:
971: /*&en
972: @table @t
973: @item Reference
974: @code{sm1.start}, @code{sm1.gb}, @code{longname}
975: @end table
976: */
977: /*&ja
978: @table @t
979: @item $B;2>H(B
980: @code{sm1.start}, @code{sm1.gb}, @code{longname}
981: @end table
982: */
983:
984:
985: /*&en
986: @c sort-sm1.genericAnn
987: @node sm1.genericAnn,,, SM1 Functions
988: @subsection @code{sm1.genericAnn}
989: @findex sm1.genericAnn
990: @table @t
991: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
992: :: It computes the annihilating ideal for @var{f}^s.
993: @var{v} is the list of variables. Here, s is @var{v}[0] and
994: @var{f} is a polynomial in the variables @code{rest}(@var{v}).
995: @end table
996:
997: @table @var
998: @item return
999: List
1000: @item p
1001: Number
1002: @item f
1003: Polynomial
1004: @item v
1005: List
1006: @end table
1007:
1008: @itemize @bullet
1009: @item This function computes the annihilating ideal for @var{f}^s.
1010: @var{v} is the list of variables. Here, s is @var{v}[0] and
1011: @var{f} is a polynomial in the variables @code{rest}(@var{v}).
1012: @end itemize
1013: */
1014: /*&ja
1015: @c sort-sm1.genericAnn
1016: @node sm1.genericAnn,,, SM1 Functions
1017: @subsection @code{sm1.genericAnn}
1018: @findex sm1.genericAnn
1019: @table @t
1020: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
1021: :: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
1022: @var{v} $B$OJQ?t$N%j%9%H$G$"$k(B. $B$3$3$G(B, s $B$O(B @var{v}[0] $B$G$"$j(B,
1023: @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
1024: @end table
1025:
1026: @table @var
1027: @item return
1028: $B%j%9%H(B
1029: @item p
1030: $B?t(B
1031: @item f
1032: $BB?9`<0(B
1033: @item v
1034: $B%j%9%H(B
1035: @end table
1036:
1037: @itemize @bullet
1038: @item $B$3$NH!?t$O(B,
1039: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
1040: @var{v} $B$OJQ?t$N%j%9%H$G$"$k(B. $B$3$3$G(B, s $B$O(B @var{v}[0] $B$G$"$j(B,
1041: @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
1042: @end itemize
1043: */
1044: /*&C
1045: @example
1046: [595] sm1.genericAnn([x^3+y^3+z^3,[s,x,y,z]]);
1047: [-x*dx-y*dy-z*dz+3*s,z^2*dy-y^2*dz,z^2*dx-x^2*dz,y^2*dx-x^2*dy]
1048: @end example
1049: */
1050: /*&en
1051: @table @t
1052: @item Reference
1053: @code{sm1.start}
1054: @end table
1055: */
1056: /*&ja
1057: @table @t
1058: @item $B;2>H(B
1059: @code{sm1.start}
1060: @end table
1061: */
1062:
1063:
1064:
1065: /*&en
1066: @c sort-sm1.wTensor0
1067: @node sm1.wTensor0,,, SM1 Functions
1068: @subsection @code{sm1.wTensor0}
1069: @findex sm1.wTensor0
1070: @table @t
1071: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1072: :: It computes the D-module theoretic 0-th tensor product
1073: of @var{f} and @var{g}.
1074: @end table
1075:
1076: @table @var
1077: @item return
1078: List
1079: @item p
1080: Number
1081: @item f, g, v, w
1082: List
1083: @end table
1084:
1085: @itemize @bullet
1086: @item
1087: It returns the D-module theoretic 0-th tensor product
1088: of @var{f} and @var{g}.
1089: @item
1090: @var{v} is a list of variables.
1091: @var{w} is a list of weights. The integer @var{w}[i] is
1092: the weight of the variable @var{v}[i].
1093: @item
1094: @code{sm1.wTensor0} calls @code{wRestriction0} of @code{ox_sm1},
1095: which requires a generic weight
1096: vector @var{w} to compute the restriction.
1097: If @var{w} is not generic, the computation fails.
1098: @item Let F and G be solutions of @var{f} and @var{g} respectively.
1099: Intuitively speaking, the 0-th tensor product is a system of
1100: differential equations which annihilates the function FG.
1101: @item The answer is a submodule of a free module D^r in general even if
1102: the inputs @var{f} and @var{g} are left ideals of D.
1103: @end itemize
1104: */
1105:
1106: /*&ja
1107: @c sort-sm1.wTensor0
1108: @node sm1.wTensor0,,, SM1 Functions
1109: @subsection @code{sm1.wTensor0}
1110: @findex sm1.wTensor0
1111: @table @t
1112: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1113: :: @var{f} $B$H(B @var{g} $B$N(B D-module $B$H$7$F$N(B 0 $B<!%F%s%=%k@Q$r(B
1114: $B7W;;$9$k(B.
1115: @end table
1116:
1117: @table @var
1118: @item return
1119: $B%j%9%H(B
1120: @item p
1121: $B?t(B
1122: @item f, g, v, w
1123: $B%j%9%H(B
1124: @end table
1125:
1126: @itemize @bullet
1127: @item
1128: @var{f} $B$H(B @var{g} $B$N(B
1129: D-$B2C72$H$7$F$N(B 0 $B<!%F%s%=%k@Q$r7W;;$9$k(B.
1130: @item
1131: @var{v} $B$OJQ?t$N%j%9%H$G$"$k(B.
1132: @var{w} $B$O(B weight $B$N%j%9%H$G$"$k(B.
1133: $B@0?t(B @var{w}[i] $B$OJQ?t(B @var{v}[i] $B$N(B weight $B$G$"$k(B.
1134: @item
1135: @code{sm1.wTensor0} $B$O(B @code{ox_sm1} $B$N(B @code{wRestriction0}
1136: $B$r$h$s$G$$$k(B.
1137: @code{wRestriction0} $B$O(B, generic $B$J(B weight $B%Y%/%H%k(B @var{w}
1138: $B$r$b$H$K$7$F@)8B$r7W;;$7$F$$$k(B.
1139: Weight $B%Y%/%H%k(B @var{w} $B$,(B generic $B$G$J$$$H7W;;$,%(%i!<$GDd;_$9$k(B.
1140: @item F $B$*$h$S(B G $B$r(B @var{f} $B$H(B @var{g} $B$=$l$>$l$N2r$H$9$k(B.
1141: $BD>4QE*$K$$$($P(B, 0 $B<!$N%F%s%=%k@Q$O(B $B4X?t(B FG $B$N$_$?$9HyJ,J}Dx<07O$G$"$k(B.
1142: @item $BF~NO(B @var{f}, @var{g} $B$,(B D $B$N:8%$%G%"%k$G$"$C$F$b(B,
1143: $B0lHL$K(B, $B=PNO$O<+M32C72(B D^r $B$NItJ,2C72$G$"$k(B.
1144: @end itemize
1145: */
1146: /*&C
1147: @example
1148: [258] sm1.wTensor0([[x*dx -1, y*dy -4],[dx+dy,dx-dy^2],[x,y],[1,2]]);
1149: [[-y*x*dx-y*x*dy+4*x+y],[5*x*dx^2+5*x*dx+2*y*dy^2+(-2*y-6)*dy+3],
1150: [-25*x*dx+(-5*y*x-2*y^2)*dy^2+((5*y+15)*x+2*y^2+16*y)*dy-20*x-8*y-15],
1151: [y^2*dy^2+(-y^2-8*y)*dy+4*y+20]]
1152: @end example
1153: */
1154:
1155:
1156:
1157: /*&en
1158: @c sort-sm1.reduction
1159: @node sm1.reduction,,, SM1 Functions
1160: @subsection @code{sm1.reduction}
1161: @findex sm1.reduction
1162: @table @t
1163: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1164: ::
1165: @end table
1166:
1167: @table @var
1168: @item return
1169: List
1170: @item f
1171: Polynomial
1172: @item g, v, w
1173: List
1174: @item p
1175: Number (the process number of ox_sm1)
1176: @end table
1177:
1178: @itemize @bullet
1179: @item It reduces @var{f} by the set of polynomial @var{g}
1180: in the homogenized Weyl algebra; it applies the
1181: division algorithm to @var{f}. The set of variables is @var{v} and
1182: @var{w} is weight vectors to determine the order, which can be ommited.
1183: @code{sm1.reduction_noH} is for the Weyl algebra.
1184: @item The return value is of the form
1185: [r,c0,[c1,...,cm],[g1,...gm]] where @var{g}=[g1, ..., gm] and
1186: c0 f + c1 g1 + ... + cm gm = r.
1187: r/c0 is the normal form.
1188: @item The function reduction reduces reducible terms that appear
1189: in lower order terms.
1190: @item The functions
1191: sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G)
1192: are for distributed polynomials.
1193: @end itemize
1194: */
1195: /*&ja
1196: @node sm1.reduction,,, SM1 Functions
1197: @subsection @code{sm1.reduction}
1198: @findex sm1.reduction
1199: @table @t
1200: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1201: ::
1202: @end table
1203:
1204: @table @var
1205: @item return
1206: $B%j%9%H(B
1207: @item f
1208: $BB?9`<0(B
1209: @item g, v, w
1210: $B%j%9%H(B
1211: @item p
1212: $B?t(B (ox_sm1 $B$N%W%m%;%9HV9f(B)
1213: @end table
1214:
1215: @itemize @bullet
1216: @item $B$3$NH!?t$O(B @var{f} $B$r(B homogenized $B%o%$%kBe?t$K$*$$$F(B,
1217: $BB?9`<0=89g(B @var{g} $B$G4JC12=(B (reduce) $B$9$k(B; $B$D$^$j(B,
1218: $B$3$NH!?t$O(B, @var{f} $B$K3d;;%"%k%4%j%:%`$rE,MQ$9$k(B.
1219: $BJQ?t=89g$O(B @var{v} $B$G;XDj$9$k(B.
1220: @var{w} $B$O=g=x$r;XDj$9$k$?$a$N(B $B%&%(%$%H%Y%/%H%k$G$"$j(B,
1221: $B>JN,$7$F$b$h$$(B.
1222: @code{sm1.reduction_noH} $B$O(B, Weyl algebra $BMQ(B.
1223: @item $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
1224: [r,c0,[c1,...,cm],g] $B$3$3$G(B @var{g}=[g1, ..., gm] $B$G$"$j(B,
1225: c0 f + c1 g1 + ... + cm gm = r
1226: $B$,$J$j$?$D(B.
1227: r/c0 $B$,(B normal form $B$G$"$k(B.
1228: @item $B$3$NH!?t$O(B, $BDc<!9`$K$"$i$o$l$k(B reducible $B$J9`$b4JC12=$9$k(B.
1229: @item $BH!?t(B
1230: sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_d(P,F,G)
1231: $B$O(B, $BJ,;6B?9`<0MQ$G$"$k(B.
1232: @end itemize
1233: */
1234: /*&C
1235: @example
1236: [259] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y]]);
1237: [x^2+y^2-4,1,[0,0],[y^4-4*y^2+1,x+y^3-4*y]]
1238: [260] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y],[[x,1]]]);
1239: [0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]]
1240: @end example
1241: */
1242: /*&en
1243: @table @t
1244: @item Reference
1245: @code{sm1.start}, @code{d_true_nf}
1246: @end table
1247: */
1248: /*&ja
1249: @table @t
1250: @item $B;2>H(B
1251: @code{sm1.start}, @code{d_true_nf}
1252: @end table
1253: */
1254:
1255:
1256: /*&en
1257: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
1258: @subsection @code{sm1.xml_tree_to_prefix_string}
1259: @findex sm1.xml_tree_to_prefix_string
1260: @table @t
1261: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1262: :: Translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
1263: @end table
1264:
1265: @table @var
1266: @item return
1267: String
1268: @item p
1269: Number
1270: @item s
1271: String
1272: @end table
1273:
1274: @itemize @bullet
1275: @item It translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
1276: @item This function should be moved to om_* in a future.
1277: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} returns CMO_TREE.
1278: asir has not yet understood this CMO.
1279: @item @code{java} execution environment is required.
1280: (For example, @code{/usr/local/jdk1.1.8/bin} should be in the
1281: command search path.)
1282: @end itemize
1283: */
1284: /*&ja
1285: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
1286: @subsection @code{sm1.xml_tree_to_prefix_string}
1287: @findex sm1.xml_tree_to_prefix_string
1288: @table @t
1289: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1290: :: XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
1291: @end table
1292:
1293: @table @var
1294: @item return
1295: String
1296: @item p
1297: Number
1298: @item s
1299: String
1300: @end table
1301:
1302: @itemize @bullet
1303: @item XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
1304: @item $B$3$NH!?t$O(B om_* $B$K>-Mh0\$9$Y$-$G$"$k(B.
1305: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} $B$O(B CMO_TREE
1306: $B$rLa$9(B. @code{asir} $B$O$3$N(B CMO $B$r$^$@%5%]!<%H$7$F$$$J$$(B.
1307: @item @code{java} $B$N<B9T4D6-$,I,MW(B.
1308: ($B$?$H$($P(B, /usr/local/jdk1.1.8/bin $B$r%3%^%s%I%5!<%A%Q%9$KF~$l$k$J$I(B.)
1309: @end itemize
1310: */
1311: /*&C
1312: @example
1313: [263] load("om");
1314: 1
1315: [270] F=om_xml(x^4-1);
1316: control: wait OX
1317: Trying to connect to the server... Done.
1318: <OMOBJ><OMA><OMS name="plus" cd="basic"/><OMA>
1319: <OMS name="times" cd="basic"/><OMA>
1320: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>4</OMI></OMA>
1321: <OMI>1</OMI></OMA><OMA><OMS name="times" cd="basic"/><OMA>
1322: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>0</OMI></OMA>
1323: <OMI>-1</OMI></OMA></OMA></OMOBJ>
1324: [271] sm1.xml_tree_to_prefix_string(F);
1325: basic_plus(basic_times(basic_power(x,4),1),basic_times(basic_power(x,0),-1))
1326: @end example
1327: */
1328: /*&en
1329: @table @t
1330: @item Reference
1331: @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
1332: @end table
1333: */
1334: /*&ja
1335: @table @t
1336: @item $B;2>H(B
1337: @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
1338: @end table
1339: */
1340:
1341:
1342:
1343:
1344: /*&en
1345: @c sort-sm1.syz
1346: @node sm1.syz,,, SM1 Functions
1347: @subsection @code{sm1.syz}
1348: @findex sm1.syz
1349: @findex sm1.syz_d
1350: @table @t
1351: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1352: :: computes the syzygy of @var{f} in the ring of differential
1353: operators with the variable @var{v}.
1354: @end table
1355:
1356: @table @var
1357: @item return
1358: List
1359: @item p
1360: Number
1361: @item f, v, w
1362: List
1363: @end table
1364:
1365: @itemize @bullet
1366: @item
1367: The return values is of the form
1368: [@var{s},[@var{g}, @var{m}, @var{t}]].
1369: Here @var{s} is the syzygy of @var{f} in the ring of differential
1370: operators with the variable @var{v}.
1371: @var{g} is a Groebner basis of @var{f} with the weight vector @var{w},
1372: and @var{m} is a matrix that translates the input matrix @var{f} to the Gr\"obner
1373: basis @var{g}.
1374: @var{t} is the syzygy of the Gr\"obner basis @var{g}.
1375: In summary, @var{g} = @var{m} @var{f} and
1376: @var{s} @var{f} = 0 hold as matrices.
1377: @item
1378: The weight vectors are given by @var{w}, which can be omitted.
1379: If @var{w} is not given,
1380: the graded reverse lexicographic order will be used to compute Grobner basis.
1381: @item
1382: When a non-term order is given, the Grobner basis is computed in
1383: the homogenized Weyl algebra (See Section 1.2 of the book of SST).
1384: The homogenization variable h is automatically added.
1385: @end itemize
1386: */
1387: /*&ja
1388: @c sort-sm1.syz
1389: @node sm1.syz,,, SM1 Functions
1390: @subsection @code{sm1.syz}
1391: @findex sm1.syz
1392: @findex sm1.syz_d
1393: @table @t
1394: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1395: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N(B syzygy $B$r7W;;$9$k(B.
1396: @end table
1397:
1398: @table @var
1399: @item return
1400: $B%j%9%H(B
1401: @item p
1402: $B?t(B
1403: @item f, v, w
1404: $B%j%9%H(B
1405: @end table
1406:
1407: @itemize @bullet
1408: @item
1409: $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
1410: [@var{s},[@var{g}, @var{m}, @var{t}]].
1411: $B$3$3$G(B @var{s} $B$O(B @var{f} $B$N(B @var{v} $B$rJQ?t$H$9$kHyJ,:nMQAG4D$K$*$1$k(B
1412: syzygy $B$G$"$k(B.
1413: @var{g} $B$O(B @var{f} $B$N(B weight vector @var{w} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
1414: @var{m} $B$OF~NO9TNs(B @var{f} $B$r%0%l%V%J4pDl(B
1415: @var{g} $B$XJQ49$9$k9TNs$G$"$k(B.
1416: @var{t} $B$O%0%l%V%J4pDl(B @var{g} $B$N(B syzygy $B$G$"$k(B.
1417: $B$^$H$a$k$H(B, $B<!$NEy<0$,$J$j$?$D(B:
1418: @var{g} = @var{m} @var{f} ,
1419: @var{s} @var{f} = 0.
1420: @item
1421: Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
1422: $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
1423: $B%V%l%V%J4pDl$r7W;;$9$k(B.
1424: @item
1425: Term order $B$G$J$$=g=x$,M?$($i$l$?>l9g$O(B, $BF1<!2=%o%$%kBe?t$G%0%l%V%J4pDl$,7W;;$5$l$k(B (SST $B$NK\$N(B Section 1.2 $B$r8+$h(B).
1426: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
1427: @end itemize
1428: */
1429: /*&C
1430: @example
1431: [293] sm1.syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
1432: [[[y*x*dy*dx-2,-x*dx-y*dy+1]], generators of the syzygy
1433: [[[x*dx+y*dy-1],[y^2*dy^2+2]], grobner basis
1434: [[1,0],[y*dy,-1]], transformation matrix
1435: [[y*x*dy*dx-2,-x*dx-y*dy+1]]]]
1436: @end example
1437: */
1438: /*&C
1439: @example
1440: [294]sm1.syz([[x^2*dx^2+x*dx+y^2*dy^2+y*dy-4,x*y*dx*dy-1],[x,y],[[dx,-1,x,1]]]);
1441: [[[y*x*dy*dx-1,-x^2*dx^2-x*dx-y^2*dy^2-y*dy+4]], generators of the syzygy
1442: [[[x^2*dx^2+h^2*x*dx+y^2*dy^2+h^2*y*dy-4*h^4],[y*x*dy*dx-h^4], GB
1443: [h^4*x*dx+y^3*dy^3+3*h^2*y^2*dy^2-3*h^4*y*dy]],
1444: [[1,0],[0,1],[y*dy,-x*dx]], transformation matrix
1445: [[y*x*dy*dx-h^4,-x^2*dx^2-h^2*x*dx-y^2*dy^2-h^2*y*dy+4*h^4]]]]
1446: @end example
1447: */
1448:
1449:
1450:
1451: /*&en
1452: @node sm1.mul,,, SM1 Functions
1453: @subsection @code{sm1.mul}
1454: @findex sm1.mul
1455: @table @t
1456: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1457: :: ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
1458: @end table
1459:
1460: @table @var
1461: @item return
1462: Polynomial or List
1463: @item p
1464: Number
1465: @item f, g
1466: Polynomial or List
1467: @item v
1468: List
1469: @end table
1470:
1471: @itemize @bullet
1472: @item Ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
1473: @item @code{sm1.mul_h} is for homogenized Weyl algebra.
1474: @item BUG: @code{sm1.mul(p0*dp0,1,[p0])} returns
1475: @code{dp0*p0+1}.
1476: A variable order such that d-variables come after non-d-variables
1477: is necessary for the correct computation.
1478: @end itemize
1479: */
1480:
1481: /*&ja
1482: @node sm1.mul,,, SM1 Functions
1483: @subsection @code{sm1.mul}
1484: @findex sm1.mul
1485: @table @t
1486: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1487: :: sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
1488: $B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
1489: @end table
1490:
1491: @table @var
1492: @item return
1493: $BB?9`<0$^$?$O%j%9%H(B
1494: @item p
1495: $B?t(B
1496: @item f, g
1497: $BB?9`<0$^$?$O%j%9%H(B
1498: @item v
1499: $B%j%9%H(B
1500: @end table
1501:
1502: @itemize @bullet
1503: @item sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
1504: $B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
1505: @item @code{sm1.mul_h} $B$O(B homogenized Weyl $BBe?tMQ(B.
1506: @item BUG: @code{sm1.mul(p0*dp0,1,[p0])} $B$O(B
1507: @code{dp0*p0+1} $B$rLa$9(B.
1508: d$BJQ?t$,8e$m$K$/$k$h$&$JJQ?t=g=x$,$O$$$C$F$$$J$$$H(B, $B$3$N4X?t$O@5$7$$Ez$($rLa$5$J$$(B.
1509: @end itemize
1510: */
1511:
1512: /*&C
1513:
1514: @example
1515: [277] sm1.mul(dx,x,[x]);
1516: x*dx+1
1517: [278] sm1.mul([x,y],[1,2],[x,y]);
1518: x+2*y
1519: [279] sm1.mul([[1,2],[3,4]],[[x,y],[1,2]],[x,y]);
1520: [[x+2,y+4],[3*x+4,3*y+8]]
1521: @end example
1522:
1523: */
1524:
1525:
1526:
1527:
1528: /*&en
1529: @node sm1.distraction,,, SM1 Functions
1530: @subsection @code{sm1.distraction}
1531: @findex sm1.distraction
1532: @table @t
1533: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1534: :: ask the @code{sm1} server to compute the distraction of @var{f}.
1535: @end table
1536:
1537: @table @var
1538: @item return
1539: List
1540: @item p
1541: Number
1542: @item f
1543: Polynomial
1544: @item v,x,d,s
1545: List
1546: @end table
1547:
1548: @itemize @bullet
1549: @item It asks the @code{sm1} server of the descriptor number @var{p}
1550: to compute the distraction of @var{f} in the ring of differential
1551: operators with variables @var{v}.
1552: @item @var{x} is a list of x-variables and @var{d} is that of d-variables
1553: to be distracted. @var{s} is a list of variables to express the distracted @var{f}.
1554: @item Distraction is roughly speaking to replace x*dx by a single variable x.
1555: See Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations at page 68 for details.
1556: @end itemize
1557: */
1558:
1559: /*&ja
1560: @node sm1.distraction,,, SM1 Functions
1561:
1562: @subsection @code{sm1.distraction}
1563: @findex sm1.distraction
1564: @table @t
1565: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1566: :: @code{sm1} $B$K(B @var{f} $B$N(B distraction $B$r7W;;$7$F$b$i$&(B.
1567: @end table
1568:
1569: @table @var
1570: @item return
1571: $B%j%9%H(B
1572: @item p
1573: $B?t(B
1574: @item f
1575: $BB?9`<0(B
1576: @item v,x,d,s
1577: $B%j%9%H(B
1578: @end table
1579:
1580: @itemize @bullet
1581: @item $B<1JL;R(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B,
1582: @var{f} $B$N(B distraction $B$r(B @var{v} $B>e$NHyJ,:nMQAG4D$G7W;;$7$F$b$i$&(B.
1583: @item @var{x} , @var{d} $B$O(B, $B$=$l$>$l(B, distract $B$9$Y$-(B x $BJQ?t(B, d $BJQ?t$N(B
1584: $B%j%9%H(B. Distraction $B$7$?$i(B, @var{s} $B$rJQ?t$H$7$F7k2L$rI=$9(B.
1585: @item Distraction $B$H$$$&$N$O(B x*dx $B$r(B x $B$GCV$-49$($k$3$H$G$"$k(B.
1586: $B>\$7$/$O(B Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations $B$N(B page 68 $B$r8+$h(B.
1587: @end itemize
1588: */
1589:
1590: /*&C
1591:
1592: @example
1593: [280] sm1.distraction([x*dx,[x],[x],[dx],[x]]);
1594: x
1595: [281] sm1.distraction([dx^2,[x],[x],[dx],[x]]);
1596: x^2-x
1597: [282] sm1.distraction([x^2,[x],[x],[dx],[x]]);
1598: x^2+3*x+2
1599: [283] fctr(@@);
1600: [[1,1],[x+1,1],[x+2,1]]
1601: [284] sm1.distraction([x*dx*y+x^2*dx^2*dy,[x,y],[x],[dx],[x]]);
1602: (x^2-x)*dy+x*y
1603: @end example
1604: */
1605:
1606: /*&en
1607: @table @t
1608: @item Reference
1609: @code{distraction2(sm1)},
1610: @end table
1611: */
1612:
1613: /*&ja
1614: @table @t
1615: @item $B;2>H(B
1616: @code{distraction2(sm1)},
1617: @end table
1618: */
1619:
1620:
1621:
1622: /*&en
1623: @node sm1.gkz,,, SM1 Functions
1624: @subsection @code{sm1.gkz}
1625: @findex sm1.gkz
1626: @table @t
1627: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1628: :: Returns the GKZ system (A-hypergeometric system) associated to the matrix
1629: @var{A} with the parameter vector @var{B}.
1630: @end table
1631:
1632: @table @var
1633: @item return
1634: List
1635: @item p
1636: Number
1637: @item A, B
1638: List
1639: @end table
1640:
1641: @itemize @bullet
1642: @item Returns the GKZ hypergeometric system
1643: (A-hypergeometric system) associated to the matrix
1644: @end itemize
1645: */
1646:
1647: /*&ja
1648: @node sm1.gkz,,, SM1 Functions
1649: @subsection @code{sm1.gkz}
1650: @findex sm1.gkz
1651: @table @t
1652: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1653: :: $B9TNs(B @var{A} $B$H%Q%i%a!<%?(B @var{B} $B$KIU?o$7$?(B GKZ $B7O(B (A-hypergeometric system) $B$r$b$I$9(B.
1654: @end table
1655:
1656: @table @var
1657: @item return
1658: $B%j%9%H(B
1659: @item p
1660: $B?t(B
1661: @item A, B
1662: $B%j%9%H(B
1663: @end table
1664:
1665: @itemize @bullet
1666: @item $B9TNs(B @var{A} $B$H%Q%i%a!<%?(B @var{B} $B$KIU?o$7$?(B GKZ $B7O(B (A-hypergeometric system) $B$r$b$I$9(B.
1667: @end itemize
1668: */
1669:
1670: /*&C
1671:
1672: @example
1673:
1674: [280] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
1675: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1676: -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1677: [x1,x2,x3,x4]]
1678:
1679: @end example
1680:
1681: */
1682:
1.2 takayama 1683: /*&en
1684: @node sm1.mgkz,,, SM1 Functions
1685: @subsection @code{sm1.mgkz}
1686: @findex sm1.mgkz
1687: @table @t
1688: @item sm1.mgkz([@var{A},@var{W},@var{B}]|proc=@var{p})
1689: :: Returns the modified GKZ system (A-hypergeometric system) associated to the matrix
1690: @var{A} and the weight @var{w} with the parameter vector @var{B}.
1691: @end table
1692:
1693: @table @var
1694: @item return
1695: List
1696: @item p
1697: Number
1698: @item A, W, B
1699: List
1700: @end table
1701:
1702: @itemize @bullet
1703: @item Returns the modified GKZ hypergeometric system
1704: (A-hypergeometric system) associated to the matrix
1705: @item http://arxiv.org/abs/0707.0043
1706: @end itemize
1707: */
1708:
1709: /*&ja
1710: @node sm1.mgkz,,, SM1 Functions
1711: @subsection @code{sm1.mgkz}
1712: @findex sm1.mgkz
1713: @table @t
1714: @item sm1.mgkz([@var{A},@var{W},@var{B}]|proc=@var{p})
1715: :: $B9TNs(B @var{A}, weight @var{W} $B$H%Q%i%a!<%?(B @var{B} $B$KIU?o$7$?(B modified GKZ $B7O(B (A-hypergeometric system) $B$r$b$I$9(B.
1716: @end table
1717:
1718: @table @var
1719: @item return
1720: $B%j%9%H(B
1721: @item p
1722: $B?t(B
1723: @item A, W, B
1724: $B%j%9%H(B
1725: @end table
1726:
1727: @itemize @bullet
1728: @item $B9TNs(B @var{A}, weight vector @var{W} $B$H%Q%i%a!<%?(B @var{B} $B$KIU?o$7$?(B modified GKZ $B7O(B (A-hypergeometric system) $B$r$b$I$9(B.
1729: @item http://arxiv.org/abs/0707.0043
1730: @end itemize
1731: */
1732:
1733: /*&C
1734:
1735: @example
1736:
1737: [280] sm1.mgkz([ [[1,2,3]], [1,2,1], [a/2]]);
1738: [[6*x3*dx3+4*x2*dx2+2*x1*dx1-a,-x4*dx4+x3*dx3+2*x2*dx2+x1*dx1,
1739: -dx2+dx1^2,-x4^2*dx3+dx1*dx2],[x1,x2,x3,x4]]
1740:
1741: Modified A-hypergeometric system for
1742: A=(1,2,3), w=(1,2,1), beta=(a/2).
1743: @end example
1744:
1745: */
1746:
1.1 takayama 1747:
1748:
1749:
1750: /*&en
1751: @node sm1.appell1,,, SM1 Functions
1752: @subsection @code{sm1.appell1}
1753: @findex sm1.appell1
1754: @table @t
1755: @item sm1.appell1(@var{a}|proc=@var{p})
1756: :: Returns the Appell hypergeometric system F_1 or F_D.
1757: @end table
1758:
1759: @table @var
1760: @item return
1761: List
1762: @item p
1763: Number
1764: @item a
1765: List
1766: @end table
1767:
1768: @itemize @bullet
1769: @item Returns the hypergeometric system for the Lauricella function
1770: F_D(a,b1,b2,...,bn,c;x1,...,xn)
1771: where @var{a} =(a,c,b1,...,bn).
1772: When n=2, the Lauricella function is called the Appell function F_1.
1773: The parameters a, c, b1, ..., bn may be rational numbers.
1774: @item It does not call sm1 function appell1. As a concequence,
1775: when parameters are rational or symbolic, this function also works
1776: as well as integral parameters.
1777: @end itemize
1778: */
1779:
1780: /*&ja
1781: @node sm1.appell1,,, SM1 Functions
1782: @subsection @code{sm1.appell1}
1783: @findex sm1.appell1
1784: @table @t
1785: @item sm1.appell1(@var{a}|proc=@var{p})
1786: :: F_1 $B$^$?$O(B F_D $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
1787: @end table
1788:
1789: @table @var
1790: @item return
1791: $B%j%9%H(B
1792: @item p
1793: $B?t(B
1794: @item a
1795: $B%j%9%H(B
1796: @end table
1797:
1798: @itemize @bullet
1799: @item Appell $B$N4X?t(B F_1 $B$*$h$S(B $B$=$N(B n $BJQ?t2=$G$"$k(B Lauricella $B$N4X?t(B
1800: F_D(a,b1,b2,...,bn,c;x1,...,xn)
1801: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
1802: @var{a} =(a,c,b1,...,bn).
1803: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
1804: @item sm1 $B$N4X?t(B appell1 $B$r$h$V$o$1$G$J$$$N$G(B, $B%Q%i%a!<%?$,M-M}?t$dJ8;z<0$N>l9g$b(B
1805: $B@5$7$/F0$/(B.
1806: @end itemize
1807: */
1808:
1809: /*&C
1810:
1811: @example
1812:
1813: [281] sm1.appell1([1,2,3,4]);
1814: [[((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2+(-5*x1+2)*dx1-3,
1815: (-x2^2+x2)*dx2^2+((-x1*x2+x1)*dx1-6*x2+2)*dx2-4*x1*dx1-4,
1816: ((-x2+x1)*dx1+3)*dx2-4*dx1], equations
1817: [x1,x2]] the list of variables
1818:
1819: [282] sm1.gb(@@);
1820: [[((-x2+x1)*dx1+3)*dx2-4*dx1,((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2
1821: +(-5*x1+2)*dx1-3,(-x2^2+x2)*dx2^2+((-x2^2+x1)*dx1-3*x2+2)*dx2
1822: +(-4*x2-4*x1)*dx1-4,
1823: (x2^3+(-x1-1)*x2^2+x1*x2)*dx2^2+((-x1*x2+x1^2)*dx1+6*x2^2
1824: +(-3*x1-2)*x2+2*x1)*dx2-4*x1^2*dx1+4*x2-4*x1],
1825: [x1*dx1*dx2,-x1^2*dx1^2,-x2^2*dx1*dx2,-x1*x2^2*dx2^2]]
1826:
1827: [283] sm1.rank(sm1.appell1([1/2,3,5,-1/3]));
1828: 3
1829:
1830: [285] Mu=2$ Beta = 1/3$
1831: [287] sm1.rank(sm1.appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));
1832: 4
1833:
1834:
1835: @end example
1836:
1837: */
1838:
1839: /*&en
1840: @node sm1.appell4,,, SM1 Functions
1841: @subsection @code{sm1.appell4}
1842: @findex sm1.appell4
1843: @table @t
1844: @item sm1.appell4(@var{a}|proc=@var{p})
1845: :: Returns the Appell hypergeometric system F_4 or F_C.
1846: @end table
1847:
1848: @table @var
1849: @item return
1850: List
1851: @item p
1852: Number
1853: @item a
1854: List
1855: @end table
1856:
1857: @itemize @bullet
1858: @item Returns the hypergeometric system for the Lauricella function
1859: F_4(a,b,c1,c2,...,cn;x1,...,xn)
1860: where @var{a} =(a,b,c1,...,cn).
1861: When n=2, the Lauricella function is called the Appell function F_4.
1862: The parameters a, b, c1, ..., cn may be rational numbers.
1863: @item @item It does not call sm1 function appell4. As a concequence,
1864: when parameters are rational or symbolic, this function also works
1865: as well as integral parameters.
1866: @end itemize
1867: */
1868:
1869: /*&ja
1870: @node sm1.appell4,,, SM1 Functions
1871: @subsection @code{sm1.appell4}
1872: @findex sm1.appell4
1873: @table @t
1874: @item sm1.appell4(@var{a}|proc=@var{p})
1875: :: F_4 $B$^$?$O(B F_C $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
1876: @end table
1877:
1878: @table @var
1879: @item return
1880: $B%j%9%H(B
1881: @item p
1882: $B?t(B
1883: @item a
1884: $B%j%9%H(B
1885: @end table
1886:
1887: @itemize @bullet
1888: @item Appell $B$N4X?t(B F_4 $B$*$h$S(B $B$=$N(B n $BJQ?t2=$G$"$k(B Lauricella $B$N4X?t(B
1889: F_C(a,b,c1,c2,...,cn;x1,...,xn)
1890: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
1891: @var{a} =(a,b,c1,...,cn).
1892: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
1893: @item sm1 $B$N4X?t(B appell1 $B$r$h$V$o$1$G$J$$$N$G(B, $B%Q%i%a!<%?$,M-M}?t$dJ8;z<0$N>l9g$b(B
1894: $B@5$7$/F0$/(B.
1895: @end itemize
1896: */
1897:
1898: /*&C
1899:
1900: @example
1901:
1902: [281] sm1.appell4([1,2,3,4]);
1903: [[-x2^2*dx2^2+(-2*x1*x2*dx1-4*x2)*dx2+(-x1^2+x1)*dx1^2+(-4*x1+3)*dx1-2,
1904: (-x2^2+x2)*dx2^2+(-2*x1*x2*dx1-4*x2+4)*dx2-x1^2*dx1^2-4*x1*dx1-2],
1905: equations
1906: [x1,x2]] the list of variables
1907:
1908: [282] sm1.rank(@@);
1909: 4
1910:
1911: @end example
1912:
1913: */
1914:
1915:
1916:
1917:
1918: /*&en
1919: @node sm1.rank,,, SM1 Functions
1920: @subsection @code{sm1.rank}
1921: @findex sm1.rank
1922: @table @t
1923: @item sm1.rank(@var{a}|proc=@var{p})
1924: :: Returns the holonomic rank of the system of differential equations @var{a}.
1925: @end table
1926:
1927: @table @var
1928: @item return
1929: Number
1930: @item p
1931: Number
1932: @item a
1933: List
1934: @end table
1935:
1936: @itemize @bullet
1937: @item It evaluates the dimension of the space of holomorphic solutions
1938: at a generic point of the system of differential equations @var{a}.
1939: The dimension is called the holonomic rank.
1940: @item @var{a} is a list consisting of a list of differential equations and
1941: a list of variables.
1942: @item @code{sm1.rrank} returns the holonomic rank when @var{a} is regular
1943: holonomic. It is generally faster than @code{sm1.rank}.
1944: @end itemize
1945: */
1946:
1947: /*&ja
1948: @node sm1.rank,,, SM1 Functions
1949: @subsection @code{sm1.rank}
1950: @findex sm1.rank
1951: @table @t
1952: @item sm1.rank(@var{a}|proc=@var{p})
1953: :: $BHyJ,J}Dx<07O(B @var{a} $B$N(B holonomic rank $B$rLa$9(B.
1954: @end table
1955:
1956: @table @var
1957: @item return
1958: $B?t(B
1959: @item p
1960: $B?t(B
1961: @item a
1962: $B%j%9%H(B
1963: @end table
1964:
1965: @itemize @bullet
1966: @item $BHyJ,J}Dx<07O(B @var{a} $B$N(B, generic point $B$G$N@5B'2r$N<!85$r(B
1967: $BLa$9(B. $B$3$N<!85$r(B, holonomic rank $B$H8F$V(B.
1968: @item @var{a} $B$OHyJ,:nMQAG$N%j%9%H$HJQ?t$N%j%9%H$h$j$J$k(B.
1969: @item @var{a} $B$,(B regular holonomic $B$N$H$-$O(B @code{sm1.rrank}
1970: $B$b(B holonomic rank $B$rLa$9(B.
1971: $B$$$C$Q$s$K$3$N4X?t$NJ}$,(B @code{sm1.rank} $B$h$jAa$$(B.
1972: @end itemize
1973: */
1974:
1975: /*&C
1976:
1977: @example
1978:
1979: [284] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
1980: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1981: -dx1*dx4+dx2*dx3, -dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1982: [x1,x2,x3,x4]]
1983: [285] sm1.rrank(@@);
1984: 4
1985:
1986: [286] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [1,2]]);
1987: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1988: -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1989: [x1,x2,x3,x4]]
1990: [287] sm1.rrank(@@);
1991: 5
1992:
1993: @end example
1994:
1995: */
1996:
1997:
1998: /*&en
1999: @node sm1.auto_reduce,,, SM1 Functions
2000: @subsection @code{sm1.auto_reduce}
2001: @findex sm1.auto_reduce
2002: @table @t
2003: @item sm1.auto_reduce(@var{s}|proc=@var{p})
2004: :: Set the flag "AutoReduce" to @var{s}.
2005: @end table
2006:
2007: @table @var
2008: @item return
2009: Number
2010: @item p
2011: Number
2012: @item s
2013: Number
2014: @end table
2015:
2016: @itemize @bullet
2017: @item If @var{s} is 1, then all Grobner bases to be computed
2018: will be the reduced Grobner bases.
2019: @item If @var{s} is 0, then Grobner bases are not necessarily the reduced
2020: Grobner bases. This is the default.
2021: @end itemize
2022: */
2023:
2024: /*&ja
2025: @node sm1.auto_reduce,,, SM1 Functions
2026: @subsection @code{sm1.auto_reduce}
2027: @findex sm1.auto_reduce
2028: @table @t
2029: @item sm1.auto_reduce(@var{s}|proc=@var{p})
2030: :: $B%U%i%0(B "AutoReduce" $B$r(B @var{s} $B$K@_Dj(B.
2031: @end table
2032:
2033: @table @var
2034: @item $BLa$jCM(B
2035: $B?t(B
2036: @item p
2037: $B?t(B
2038: @item s
2039: $B?t(B
2040: @end table
2041:
2042: @itemize @bullet
2043: @item @var{s} $B$,(B 1 $B$N$H$-(B, $B0J8e7W;;$5$l$k%0%l%V%J4pDl$O$9$Y$F(B,
2044: reduced $B%0%l%V%J4pDl$H$J$k(B.
2045: @item @var{s} $B$,(B 0 $B$N$H$-(B, $B7W;;$5$l$k%0%l%V%J4pDl$O(B
2046: reduced $B%0%l%V%J4pDl$H$O$+$.$i$J$$(B. $B$3$A$i$,%G%U%)!<%k%H(B.
2047: @end itemize
2048: */
2049:
2050:
2051:
2052: /*&en
2053: @node sm1.slope,,, SM1 Functions
2054: @subsection @code{sm1.slope}
2055: @findex sm1.slope
2056: @table @t
2057: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
2058: :: Returns the slopes of differential equations @var{ii}.
2059: @end table
2060:
2061: @table @var
2062: @item return
2063: List
2064: @item p
2065: Number
2066: @item ii
2067: List (equations)
2068: @item v
2069: List (variables)
2070: @item f_filtration
2071: List (weight vector)
2072: @item v_filtration
2073: List (weight vector)
2074: @end table
2075:
2076: @itemize @bullet
2077: @item @code{sm1.slope} returns the (geometric) slopes
2078: of the system of differential equations @var{ii}
2079: along the hyperplane specified by
2080: the V filtration @var{v_filtration}.
2081: @item @var{v} is a list of variables.
2082: @item The return value is a list of lists.
2083: The first entry of each list is the slope and the second entry
2084: is the weight vector for which the microcharacteristic variety is
2085: not bihomogeneous.
2086: @end itemize
2087:
2088: @noindent
2089: Algorithm:
2090: see "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
2091: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
1.4 takayama 2092: Note that the signs of the slopes s' are negative, but the absolute values -s'
1.1 takayama 2093: of the slopes are returned.
1.4 takayama 2094: In other words, when pF+qV is the gap, -s'=q/p is returned.
2095: Note that s=1-1/s' is called the slope in recent literatures. Solutions belongs to O(s).
2096: The number s satisfies 1<= s.
2097: We have r=s-1=-1/s', and kappa=1/r=-s', which is used 1/Gamma(1+m*r) factor and exp(-tau^kappa)
2098: in the Borel and Laplace transformations respectively.
1.1 takayama 2099:
2100: */
2101:
2102: /*&ja
2103: @node sm1.slope,,, SM1 Functions
2104: @subsection @code{sm1.slope}
2105: @findex sm1.slope
2106: @table @t
2107: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
2108: :: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B slope $B$rLa$9(B.
2109: @end table
2110:
2111: @table @var
2112: @item return
2113: $B?t(B
2114: @item p
2115: $B?t(B
2116: @item ii
2117: $B%j%9%H(B ($BJ}Dx<0(B)
2118: @item v
2119: $B%j%9%H(B ($BJQ?t(B)
2120: @item f_filtration
2121: $B%j%9%H(B (weight vector)
2122: @item v_filtration
2123: $B%j%9%H(B (weight vector)
2124: @end table
2125:
2126: @itemize @bullet
2127: @item @code{sm1.slope} $B$O(B
2128: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B V filtration @var{v_filtration}
2129: $B$G;XDj$9$kD6J?LL$K1h$C$F$N(B (geomeric) slope $B$r7W;;$9$k(B.
2130: @item @var{v} $B$OJQ?t$N%j%9%H(B.
2131: @item $BLa$jCM$O(B, $B%j%9%H$r@.J,$H$9$k%j%9%H$G$"$k(B.
2132: $B@.J,%j%9%H$NBh(B 1 $BMWAG$,(B slope, $BBh(B 2 $BMWAG$O(B, $B$=$N(B weight vector $B$KBP1~$9$k(B
2133: microcharacteristic variety $B$,(B bihomogeneous $B$G$J$$(B.
2134: @end itemize
2135:
2136: @noindent
2137: Algorithm:
2138: "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
2139: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
2140: $B$r$_$h(B.
1.4 takayama 2141: Slope s' $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B, $B$3$N%W%m%0%i%`$O(B,
2142: Slope $B$N@dBPCM(B -s' $B$rLa$9(B.
2143: $B$D$^$j(B pF+qV $B$,(Bmicro$BFC@-B?MMBN$N(Bgap$B$G$"$k$H$-(B, -s'=q/p $B$rLa$9(B.
2144: $B:G6a$NJ88%$G$O(B s=1-1/s' $B$r(B slope $B$H8F$s$G$$$k(B. $B2r$O(B O(s) $B$KB0$9$k(B.
2145: $B?t(B s $B$O(B 1<= s $B$rK~$9(B.
2146: r=s-1=-1/s' $B$*$h$S(B kappa=1/r=-s' $B$G$"$k(B.
2147: $B$3$l$i$N?t$O(BBorel and Laplace $BJQ49$K$*$$$F$=$l$>$l(B 1/Gamma(1+m*r) factor,
2148: exp(-tau^kappa) $B9`$H$7$F;H$o$l$k(B.
1.1 takayama 2149: */
2150:
2151: /*&C
2152:
2153: @example
2154:
2155: [284] A= sm1.gkz([ [[1,2,3]], [-3] ]);
2156:
2157:
2158: [285] sm1.slope(A[0],A[1],[0,0,0,1,1,1],[0,0,-1,0,0,1]);
2159:
2160: [286] A2 = sm1.gkz([ [[1,1,1,0],[2,-3,1,-3]], [1,0]]);
2161: (* This is an interesting example given by Laura Matusevich,
2162: June 9, 2001 *)
2163:
2164: [287] sm1.slope(A2[0],A2[1],[0,0,0,0,1,1,1,1],[0,0,0,-1,0,0,0,1]);
2165:
2166:
2167: @end example
2168:
2169: */
2170: /*&en
2171: @table @t
2172: @item Reference
2173: @code{sm.gb}
2174: @end table
2175: */
2176: /*&ja
2177: @table @t
2178: @item $B;2>H(B
2179: @code{sm.gb}
2180: @end table
2181: */
2182:
2183:
2184: /*&en
2185: @include sm1-auto.en
2186: */
2187:
2188: /*&ja
2189: @include sm1-auto.ja
2190: */
2191:
2192:
2193: end$
2194:
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