Annotation of OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw, Revision 1.4
1.4 ! takayama 1: /*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v 1.3 2010/02/06 00:50:32 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
514: @item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
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).
562: @end itemize
563: */
564: /*&ja
565: @c sort-sm1.gb
566: @node sm1.gb,,, SM1 Functions
567: @subsection @code{sm1.gb}
568: @findex sm1.gb
569: @findex sm1.gb_d
570: @table @t
571: @item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
572: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
573: @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
574: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$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.
575: @end table
576:
577: @table @var
578: @item return
579: $B%j%9%H(B
580: @item p, q, r
581: $B?t(B
582: @item f, v, w
583: $B%j%9%H(B
584: @end table
585:
586: @itemize @bullet
587: @item
588: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
589: @item
590: Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
591: $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
592: $B%V%l%V%J4pDl$r7W;;$9$k(B.
593: @item
594: @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
595: ( @var{w} $B$,$J$$$H$-(B ) $B$^$?$O(B $B%$%K%7%!%kB?9`<0(B ( @var{w} $B$,M?$($i$?$H$-(B)
596: $B$N%j%9%H$G$"$k(B.
597: @item
598: @code{sm1.gb_d} $B$O7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.
599: $BB?9`<0$NCf$K8=$l$k%b%N%_%"%k$O%0%l%V%J4pDl$r7W;;$9$k$H$-$KM?$($i$?=g=x$G%=!<%H$5$l$F$$$k(B.
600: $BLa$jCM$O(B
601: [$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]
602: $B$G$"$k(B.
603: @item
604: Term order $B$G$J$$=g=x$,M?$($i$l$?>l9g$O(B, $BF1<!2=%o%$%kBe?t$G%0%l%V%J4pDl$,7W;;$5$l$k(B (SST $B$NK\$N(B Section 1.2 $B$r8+$h(B).
605: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
606: @item $B%*%W%7%g%J%kJQ?t(B @var{q} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
607: 3 $BHVL\$NLa$jCM$H$7$F(B, $B%0%l%V%J4pDl$*$h$S%$%K%7%!%k$N%j%9%H$,(B
608: $BM?$($i$l$?=g=x$G%=!<%H$5$l$?%b%N%_%"%k$NOB$H$7$FLa$5$l$k(B.
609: $B$$$^$N$H$3$m$3$NB?9`<0$O(B, $BJ8;zNs$GI=8=$5$l$k(B.
610: $B%*%W%7%g%J%kJQ?t(B @var{r} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
611: $BLa$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).
612: @end itemize
613: */
614: /*&C
615: @example
616: [293] sm1.gb([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
617: [[x*dx+y*dy-1,y^2*dy^2+2],[x*dx,y^2*dy^2]]
618: @end example
619: */
620: /*&en
621: In the example above,
622: @tex the set $\{ x \partial_x + y \partial_y -1,
623: y^2 \partial_y^2+2\}$
624: is the Gr\"obner basis of the input with respect to the
625: graded reverse lexicographic order such that
626: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$.
627: The set $\{x \partial_x, y^2 \partial_y\}$ is the leading monomials
628: (the initial monominals) of the Gr\"obner basis.
629: @end tex
630: */
631: /*&ja
632: $B>e$NNc$K$*$$$F(B,
633: @tex $B=89g(B $\{ x \partial_x + y \partial_y -1,
634: y^2 \partial_y^2+2\}$
635: $B$O(B
636: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$
637: $B$G$"$k$h$&$J(B
638: graded reverse lexicographic order $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
639: $B=89g(B $\{x \partial_x, y^2 \partial_y\}$ $B$O%0%l%V%J4pDl$N3F85$K(B
640: $BBP$9$k(B leading monomial (initial monomial) $B$G$"$k(B.
641: @end tex
642: */
643: /*&C
644: @example
645: [294] sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]);
646: [[dx+dy^3-4*dy,-dy^4+4*dy^2-1],[dx,-dy^4]]
647: @end example
648: */
649: /*&en
650: In the example above, two monomials
651: @tex
652: $m = x^a y^b \partial_x^c \partial_y^d$ and
653: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
654: are firstly compared by the weight vector
655: {\tt (dx,dy,x,y) = (50,2,1,0)}
656: (i.e., $m$ is larger than $m'$ if $50c+2d+a > 50c'+2d'+a'$)
657: and when the comparison is tie, then these are
658: compared by the reverse lexicographic order
659: (i.e., if $50c+2d+a = 50c'+2d'+a'$, then use the reverse lexicogrpahic order).
660: @end tex
661: */
662: /*&ja
663: $B>e$NNc$K$*$$$FFs$D$N%b%N%_%"%k(B
664: @tex
665: $m = x^a y^b \partial_x^c \partial_y^d$ $B$*$h$S(B
666: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
667: $B$O:G=i$K(B weight vector
668: {\tt (dx,dy,x,y) = (50,2,1,0)} $B$rMQ$$$FHf3S$5$l$k(B
669: ($B$D$^$j(B $m$ $B$O(B $50c+2d+a > 50c'+2d'+a'$ $B$N$H$-(B
670: $m'$ $B$h$jBg$-$$(B )
671: $B<!$K$3$NHf3S$G>!Ii$,$D$+$J$$$H$-$O(B reverse lexicographic order $B$GHf3S$5$l$k(B
672: ($B$D$^$j(B $50c+2d+a = 50c'+2d'+a'$ $B$N$H$-(B reverse lexicographic order $B$GHf3S(B
673: $B$5$l$k(B).
674: @end tex
675: */
676: /*&C
677: @example
678: [294] F=sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);
679: map(print,F[2][0])$
680: map(print,F[2][1])$
681: @end example
682: */
683: /*&C
684: @example
685: [595]
686: sm1.gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],
687: [x,y],[[dx,1,x,-1],[dy,1]]]);
688:
689: [[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],
690: [x*dx^2+(y*dy-h^2)*dx,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx]]
691:
692: [596]
693: sm1.gb_d([["dx (x dx +y dy-2)-1","dy (x dx + y dy -2)-1"],
694: "x,y",[[dx,1,x,-1],[dy,1]]]);
695: [[[e0,x,y,H,E,dx,dy,h],
696: [[0,-1,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],
697: [0,1,1,1,1,1,1,0],[0,0,0,0,0,0,-1,0],[0,0,0,0,0,-1,0,0],
698: [0,0,0,0,-1,0,0,0],[0,0,0,-1,0,0,0,0],[0,0,-1,0,0,0,0,0],
699: [0,0,0,0,0,0,0,1]]],
700: [[(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)*
701: <<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
702: ,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
703: ,1,3>>],
704: [(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)*<
705: <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
706: ,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]]
707: @end example
708: */
709:
710: /*&en
711: @table @t
712: @item Reference
713: @code{sm1.reduction}, @code{sm1.rat_to_p}
714: @end table
715: */
716: /*&ja
717: @table @t
718: @item $B;2>H(B
719: @code{sm1.reduction}, @code{sm1.rat_to_p}
720: @end table
721: */
722:
723:
724:
725: /*&en
726: @c sort-sm1.deRham
727: @node sm1.deRham,,, SM1 Functions
728: @subsection @code{sm1.deRham}
729: @findex sm1.deRham
730: @table @t
731: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
732: :: ask the server to evaluate the dimensions of the de Rham cohomology groups
733: of C^n - (the zero set of @var{f}=0).
734: @end table
735:
736: @table @var
737: @item return
738: List
739: @item p
740: Number
741: @item f
742: String or polynomial
743: @item v
744: List
745: @end table
746:
747: @itemize @bullet
748: @item It returns the dimensions of the de Rham cohomology groups
749: of X = C^n \ V(@var{f}).
750: In other words, it returns
751: [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)].
752: @item @var{v} is a list of variables. n = @code{length(@var{v})}.
753: @item
754: @code{sm1.deRham} requires huge computer resources.
755: For example, @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
756: is already very hard.
757: @item
758: To efficiently analyze the roots of b-function, @code{ox_asir} should be used
759: from @code{ox_sm1_forAsir}.
760: It is recommended to load the communication module for @code{ox_asir}
761: by the command @*
762: @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
763: This command is automatically executed when @code{ox_sm1_forAsir} is started.
764: @item If you make an interruption to the function @code{sm1.deRham}
765: by @code{ox_reset(sm1.get_Sm1_proc());}, the server might get out of the standard
766: mode. So, it is strongly recommended to execute the command
767: @code{ox_shutdown(sm1.get_Sm1_proc());} to interrupt and restart the server.
768: @end itemize
769: */
770: /*&ja
771: @c sort-sm1.deRham
772: @node sm1.deRham,,, SM1 Functions
773: @subsection @code{sm1.deRham}
774: @findex sm1.deRham
775: @table @t
776: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
777: :: $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.
778: @end table
779:
780: @table @var
781: @item return
782: $B%j%9%H(B
783: @item p
784: $B?t(B
785: @item f
786: $BJ8;zNs(B $B$^$?$O(B $BB?9`<0(B
787: @item v
788: $B%j%9%H(B
789: @end table
790:
791: @itemize @bullet
792: @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.
793: $B$9$J$o$A(B,
794: [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)]
795: $B$rLa$9(B.
796: @item @var{v} $B$OJQ?t$N%j%9%H(B. n = @code{length(@var{v})} $B$G$"$k(B.
797: @item
798: @code{sm1.deRham} $B$O7W;;5!$N;q8;$rBgNL$K;HMQ$9$k(B.
799: $B$?$H$($P(B @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
800: $B$N7W;;$9$i$9$G$KHs>o$KBgJQ$G$"$k(B.
801: @item
802: b-$B4X?t$N:,$r8zN($h$/2r@O$9$k$K$O(B, @code{ox_asir} $B$,(B @code{ox_sm1_forAsir}
803: $B$h$j;HMQ$5$l$k$Y$-$G$"$k(B. $B%3%^%s%I(B @*
804: @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
805: $B$rMQ$$$F(B, @code{ox_asir} $B$H$NDL?.%b%8%e!<%k$r$"$i$+$8$a%m!<%I$7$F$*$/$H$h$$(B.
806: $B$3$N%3%^%s%I$O(B @code{ox_asir_forAsir} $B$N%9%?!<%H;~$K<+F0E*$K<B9T$5$l$F$$$k(B.
807: @item
808: @code{sm1.deRham} $B$r(B @code{ox_reset(sm1.get_Sm1_proc());} $B$GCfCG$9$k$H(B,
809: $B0J8e(B sm1 $B%5!<%P$,HsI8=`%b!<%I$KF~$jM=4|$7$J$$F0:n$r$9$k>l9g(B
810: $B$,$"$k$N$G(B, $B%3%^%s%I(B @code{ox_shutdown(sm1.get_Sm1_proc());} $B$G(B, @code{ox_sm1_forAsir}
811: $B$r0l;~(B shutdown $B$7$F%j%9%?!<%H$7$?J}$,0BA4$G$"$k(B.
812: @end itemize
813: */
814: /*&C
815: @example
816: [332] sm1.deRham([x^3-y^2,[x,y]]);
817: [1,1,0]
818: [333] sm1.deRham([x*(x-1),[x]]);
819: [1,2]
820: @end example
821: */
822: /*&en
823: @table @t
824: @item Reference
825: @code{sm1.start}, @code{deRham} (sm1 command)
826: @item Algorithm:
827: Oaku, Takayama, An algorithm for de Rham cohomology groups of the
828: complement of an affine variety via D-module computation,
829: Journal of pure and applied algebra 139 (1999), 201--233.
830: @end table
831: */
832: /*&ja
833: @table @t
834: @item $B;2>H(B
835: @code{sm1.start}, @code{deRham} (sm1 command)
836: @item Algorithm:
837: Oaku, Takayama, An algorithm for de Rham cohomology groups of the
838: complement of an affine variety via D-module computation,
839: Journal of pure and applied algebra 139 (1999), 201--233.
840: @end table
841: */
842:
843:
844:
845:
846: /*&en
847: @c sort-sm1.hilbert
848: @node sm1.hilbert,,, SM1 Functions
849: @subsection @code{sm1.hilbert}
850: @findex sm1.hilbert
851: @findex hilbert_polynomial
852: @table @t
853: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
854: :: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
855: @item hilbert_polynomial(@var{f},@var{v})
856: :: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
857: @end table
858:
859: @table @var
860: @item return
861: Polynomial
862: @item p
863: Number
864: @item f, v
865: List
866: @end table
867:
868: @itemize @bullet
869: @item It returns the Hilbert polynomial h(k) of the set of polynomials
870: @var{f}
871: with respect to the set of variables @var{v}.
872: @item
873: h(k) = dim_Q F_k/I \cap F_k where F_k the set of polynomials of which
874: degree is less than or equal to k and I is the ideal generated by the
875: set of polynomials @var{f}.
876: @item
877: Note for sm1.hilbert:
878: For an efficient computation, it is preferable that
879: the set of polynomials @var{f} is a set of monomials.
880: In fact, this function firstly compute a Grobner basis of @var{f}, and then
881: compute the Hilbert polynomial of the initial monomials of the basis.
882: If the input @var{f} is already a Grobner
883: basis, a Grobner basis is recomputed in this function,
884: which is a waste of time and Grobner basis computation in the ring of
885: polynomials in @code{sm1} is slower than in @code{asir}.
886: @end itemize
887: */
888: /*&ja
889: @c sort-sm1.hilbert
890: @node sm1.hilbert,,, SM1 Functions
891: @subsection @code{sm1.hilbert}
892: @findex sm1.hilbert
893: @findex hilbert_polynomial
894: @table @t
895: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
896: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
897: @item hilbert_polynomial(@var{f},@var{v})
898: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
899: @end table
900:
901: @table @var
902: @item return
903: $BB?9`<0(B
904: @item p
905: $B?t(B
906: @item f, v
907: $B%j%9%H(B
908: @end table
909:
910: @itemize @bullet
911: @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)
912: $B$r7W;;$9$k(B.
913: @item
914: 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
915: $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.
916: @item
917: sm1.hilbert $B$K$+$s$9$k%N!<%H(B:
918: $B8zN($h$/7W;;$9$k$K$O(B @var{f} $B$O%b%N%_%"%k$N=89g$K$7$?J}$,$$$$(B.
919: $B<B:](B, $B$3$NH!?t$O$^$:(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$7(B, $B$=$l$+$i$=$N(B initial
920: monomial $BC#$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
921: $B$7$?$,$C$F(B, $BF~NO(B @var{f} $B$,$9$G$K%0%l%V%J4pDl$@$H$3$NH!?t$N$J$+$G$b$&0lEY(B
922: $B%0%l%V%J4pDl$N7W;;$,$*$3$J$o$l$k(B. $B$3$l$O;~4V$NL5BL$G$"$k$7(B, @code{sm1} $B$N(B
923: $BB?9`<0%0%l%V%J4pDl7W;;$O(B @code{asir} $B$h$jCY$$(B.
924: @end itemize
925: */
926:
927: /*&C
928: @example
929:
930: [346] load("katsura")$
931: [351] A=hilbert_polynomial(katsura(5),[u0,u1,u2,u3,u4,u5]);
932: 32
933:
934: @end example
935:
936: @example
937: [279] load("katsura")$
938: [280] A=gr(katsura(5),[u0,u1,u2,u3,u4,u5],0)$
939: [281] dp_ord();
940: 0
941: [282] B=map(dp_ht,map(dp_ptod,A,[u0,u1,u2,u3,u4,u5]));
942: [(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>>,
943: (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>>,
944: (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>>,
945: (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>>,
946: (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>>,
947: (1)*<<0,1,0,0,0,4>>,(1)*<<0,0,0,0,0,6>>]
948: [283] C=map(dp_dtop,B,[u0,u1,u2,u3,u4,u5]);
949: [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,
950: 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,
951: u5^4*u2,u5^4*u1,u5^6]
952: [284] sm1.hilbert([C,[u0,u1,u2,u3,u4,u5]]);
953: 32
954: @end example
955: */
956:
957: /*&en
958: @table @t
959: @item Reference
960: @code{sm1.start}, @code{sm1.gb}, @code{longname}
961: @end table
962: */
963: /*&ja
964: @table @t
965: @item $B;2>H(B
966: @code{sm1.start}, @code{sm1.gb}, @code{longname}
967: @end table
968: */
969:
970:
971: /*&en
972: @c sort-sm1.genericAnn
973: @node sm1.genericAnn,,, SM1 Functions
974: @subsection @code{sm1.genericAnn}
975: @findex sm1.genericAnn
976: @table @t
977: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
978: :: It computes the annihilating ideal for @var{f}^s.
979: @var{v} is the list of variables. Here, s is @var{v}[0] and
980: @var{f} is a polynomial in the variables @code{rest}(@var{v}).
981: @end table
982:
983: @table @var
984: @item return
985: List
986: @item p
987: Number
988: @item f
989: Polynomial
990: @item v
991: List
992: @end table
993:
994: @itemize @bullet
995: @item This function computes the annihilating ideal for @var{f}^s.
996: @var{v} is the list of variables. Here, s is @var{v}[0] and
997: @var{f} is a polynomial in the variables @code{rest}(@var{v}).
998: @end itemize
999: */
1000: /*&ja
1001: @c sort-sm1.genericAnn
1002: @node sm1.genericAnn,,, SM1 Functions
1003: @subsection @code{sm1.genericAnn}
1004: @findex sm1.genericAnn
1005: @table @t
1006: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
1007: :: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
1008: @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,
1009: @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
1010: @end table
1011:
1012: @table @var
1013: @item return
1014: $B%j%9%H(B
1015: @item p
1016: $B?t(B
1017: @item f
1018: $BB?9`<0(B
1019: @item v
1020: $B%j%9%H(B
1021: @end table
1022:
1023: @itemize @bullet
1024: @item $B$3$NH!?t$O(B,
1025: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
1026: @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,
1027: @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
1028: @end itemize
1029: */
1030: /*&C
1031: @example
1032: [595] sm1.genericAnn([x^3+y^3+z^3,[s,x,y,z]]);
1033: [-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]
1034: @end example
1035: */
1036: /*&en
1037: @table @t
1038: @item Reference
1039: @code{sm1.start}
1040: @end table
1041: */
1042: /*&ja
1043: @table @t
1044: @item $B;2>H(B
1045: @code{sm1.start}
1046: @end table
1047: */
1048:
1049:
1050:
1051: /*&en
1052: @c sort-sm1.wTensor0
1053: @node sm1.wTensor0,,, SM1 Functions
1054: @subsection @code{sm1.wTensor0}
1055: @findex sm1.wTensor0
1056: @table @t
1057: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1058: :: It computes the D-module theoretic 0-th tensor product
1059: of @var{f} and @var{g}.
1060: @end table
1061:
1062: @table @var
1063: @item return
1064: List
1065: @item p
1066: Number
1067: @item f, g, v, w
1068: List
1069: @end table
1070:
1071: @itemize @bullet
1072: @item
1073: It returns the D-module theoretic 0-th tensor product
1074: of @var{f} and @var{g}.
1075: @item
1076: @var{v} is a list of variables.
1077: @var{w} is a list of weights. The integer @var{w}[i] is
1078: the weight of the variable @var{v}[i].
1079: @item
1080: @code{sm1.wTensor0} calls @code{wRestriction0} of @code{ox_sm1},
1081: which requires a generic weight
1082: vector @var{w} to compute the restriction.
1083: If @var{w} is not generic, the computation fails.
1084: @item Let F and G be solutions of @var{f} and @var{g} respectively.
1085: Intuitively speaking, the 0-th tensor product is a system of
1086: differential equations which annihilates the function FG.
1087: @item The answer is a submodule of a free module D^r in general even if
1088: the inputs @var{f} and @var{g} are left ideals of D.
1089: @end itemize
1090: */
1091:
1092: /*&ja
1093: @c sort-sm1.wTensor0
1094: @node sm1.wTensor0,,, SM1 Functions
1095: @subsection @code{sm1.wTensor0}
1096: @findex sm1.wTensor0
1097: @table @t
1098: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1099: :: @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
1100: $B7W;;$9$k(B.
1101: @end table
1102:
1103: @table @var
1104: @item return
1105: $B%j%9%H(B
1106: @item p
1107: $B?t(B
1108: @item f, g, v, w
1109: $B%j%9%H(B
1110: @end table
1111:
1112: @itemize @bullet
1113: @item
1114: @var{f} $B$H(B @var{g} $B$N(B
1115: D-$B2C72$H$7$F$N(B 0 $B<!%F%s%=%k@Q$r7W;;$9$k(B.
1116: @item
1117: @var{v} $B$OJQ?t$N%j%9%H$G$"$k(B.
1118: @var{w} $B$O(B weight $B$N%j%9%H$G$"$k(B.
1119: $B@0?t(B @var{w}[i] $B$OJQ?t(B @var{v}[i] $B$N(B weight $B$G$"$k(B.
1120: @item
1121: @code{sm1.wTensor0} $B$O(B @code{ox_sm1} $B$N(B @code{wRestriction0}
1122: $B$r$h$s$G$$$k(B.
1123: @code{wRestriction0} $B$O(B, generic $B$J(B weight $B%Y%/%H%k(B @var{w}
1124: $B$r$b$H$K$7$F@)8B$r7W;;$7$F$$$k(B.
1125: Weight $B%Y%/%H%k(B @var{w} $B$,(B generic $B$G$J$$$H7W;;$,%(%i!<$GDd;_$9$k(B.
1126: @item F $B$*$h$S(B G $B$r(B @var{f} $B$H(B @var{g} $B$=$l$>$l$N2r$H$9$k(B.
1127: $BD>4QE*$K$$$($P(B, 0 $B<!$N%F%s%=%k@Q$O(B $B4X?t(B FG $B$N$_$?$9HyJ,J}Dx<07O$G$"$k(B.
1128: @item $BF~NO(B @var{f}, @var{g} $B$,(B D $B$N:8%$%G%"%k$G$"$C$F$b(B,
1129: $B0lHL$K(B, $B=PNO$O<+M32C72(B D^r $B$NItJ,2C72$G$"$k(B.
1130: @end itemize
1131: */
1132: /*&C
1133: @example
1134: [258] sm1.wTensor0([[x*dx -1, y*dy -4],[dx+dy,dx-dy^2],[x,y],[1,2]]);
1135: [[-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],
1136: [-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],
1137: [y^2*dy^2+(-y^2-8*y)*dy+4*y+20]]
1138: @end example
1139: */
1140:
1141:
1142:
1143: /*&en
1144: @c sort-sm1.reduction
1145: @node sm1.reduction,,, SM1 Functions
1146: @subsection @code{sm1.reduction}
1147: @findex sm1.reduction
1148: @table @t
1149: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1150: ::
1151: @end table
1152:
1153: @table @var
1154: @item return
1155: List
1156: @item f
1157: Polynomial
1158: @item g, v, w
1159: List
1160: @item p
1161: Number (the process number of ox_sm1)
1162: @end table
1163:
1164: @itemize @bullet
1165: @item It reduces @var{f} by the set of polynomial @var{g}
1166: in the homogenized Weyl algebra; it applies the
1167: division algorithm to @var{f}. The set of variables is @var{v} and
1168: @var{w} is weight vectors to determine the order, which can be ommited.
1169: @code{sm1.reduction_noH} is for the Weyl algebra.
1170: @item The return value is of the form
1171: [r,c0,[c1,...,cm],[g1,...gm]] where @var{g}=[g1, ..., gm] and
1172: c0 f + c1 g1 + ... + cm gm = r.
1173: r/c0 is the normal form.
1174: @item The function reduction reduces reducible terms that appear
1175: in lower order terms.
1176: @item The functions
1177: sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G)
1178: are for distributed polynomials.
1179: @end itemize
1180: */
1181: /*&ja
1182: @node sm1.reduction,,, SM1 Functions
1183: @subsection @code{sm1.reduction}
1184: @findex sm1.reduction
1185: @table @t
1186: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1187: ::
1188: @end table
1189:
1190: @table @var
1191: @item return
1192: $B%j%9%H(B
1193: @item f
1194: $BB?9`<0(B
1195: @item g, v, w
1196: $B%j%9%H(B
1197: @item p
1198: $B?t(B (ox_sm1 $B$N%W%m%;%9HV9f(B)
1199: @end table
1200:
1201: @itemize @bullet
1202: @item $B$3$NH!?t$O(B @var{f} $B$r(B homogenized $B%o%$%kBe?t$K$*$$$F(B,
1203: $BB?9`<0=89g(B @var{g} $B$G4JC12=(B (reduce) $B$9$k(B; $B$D$^$j(B,
1204: $B$3$NH!?t$O(B, @var{f} $B$K3d;;%"%k%4%j%:%`$rE,MQ$9$k(B.
1205: $BJQ?t=89g$O(B @var{v} $B$G;XDj$9$k(B.
1206: @var{w} $B$O=g=x$r;XDj$9$k$?$a$N(B $B%&%(%$%H%Y%/%H%k$G$"$j(B,
1207: $B>JN,$7$F$b$h$$(B.
1208: @code{sm1.reduction_noH} $B$O(B, Weyl algebra $BMQ(B.
1209: @item $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
1210: [r,c0,[c1,...,cm],g] $B$3$3$G(B @var{g}=[g1, ..., gm] $B$G$"$j(B,
1211: c0 f + c1 g1 + ... + cm gm = r
1212: $B$,$J$j$?$D(B.
1213: r/c0 $B$,(B normal form $B$G$"$k(B.
1214: @item $B$3$NH!?t$O(B, $BDc<!9`$K$"$i$o$l$k(B reducible $B$J9`$b4JC12=$9$k(B.
1215: @item $BH!?t(B
1216: sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_d(P,F,G)
1217: $B$O(B, $BJ,;6B?9`<0MQ$G$"$k(B.
1218: @end itemize
1219: */
1220: /*&C
1221: @example
1222: [259] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y]]);
1223: [x^2+y^2-4,1,[0,0],[y^4-4*y^2+1,x+y^3-4*y]]
1224: [260] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y],[[x,1]]]);
1225: [0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]]
1226: @end example
1227: */
1228: /*&en
1229: @table @t
1230: @item Reference
1231: @code{sm1.start}, @code{d_true_nf}
1232: @end table
1233: */
1234: /*&ja
1235: @table @t
1236: @item $B;2>H(B
1237: @code{sm1.start}, @code{d_true_nf}
1238: @end table
1239: */
1240:
1241:
1242: /*&en
1243: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
1244: @subsection @code{sm1.xml_tree_to_prefix_string}
1245: @findex sm1.xml_tree_to_prefix_string
1246: @table @t
1247: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1248: :: Translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
1249: @end table
1250:
1251: @table @var
1252: @item return
1253: String
1254: @item p
1255: Number
1256: @item s
1257: String
1258: @end table
1259:
1260: @itemize @bullet
1261: @item It translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
1262: @item This function should be moved to om_* in a future.
1263: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} returns CMO_TREE.
1264: asir has not yet understood this CMO.
1265: @item @code{java} execution environment is required.
1266: (For example, @code{/usr/local/jdk1.1.8/bin} should be in the
1267: command search path.)
1268: @end itemize
1269: */
1270: /*&ja
1271: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
1272: @subsection @code{sm1.xml_tree_to_prefix_string}
1273: @findex sm1.xml_tree_to_prefix_string
1274: @table @t
1275: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1276: :: XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
1277: @end table
1278:
1279: @table @var
1280: @item return
1281: String
1282: @item p
1283: Number
1284: @item s
1285: String
1286: @end table
1287:
1288: @itemize @bullet
1289: @item XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
1290: @item $B$3$NH!?t$O(B om_* $B$K>-Mh0\$9$Y$-$G$"$k(B.
1291: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} $B$O(B CMO_TREE
1292: $B$rLa$9(B. @code{asir} $B$O$3$N(B CMO $B$r$^$@%5%]!<%H$7$F$$$J$$(B.
1293: @item @code{java} $B$N<B9T4D6-$,I,MW(B.
1294: ($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.)
1295: @end itemize
1296: */
1297: /*&C
1298: @example
1299: [263] load("om");
1300: 1
1301: [270] F=om_xml(x^4-1);
1302: control: wait OX
1303: Trying to connect to the server... Done.
1304: <OMOBJ><OMA><OMS name="plus" cd="basic"/><OMA>
1305: <OMS name="times" cd="basic"/><OMA>
1306: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>4</OMI></OMA>
1307: <OMI>1</OMI></OMA><OMA><OMS name="times" cd="basic"/><OMA>
1308: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>0</OMI></OMA>
1309: <OMI>-1</OMI></OMA></OMA></OMOBJ>
1310: [271] sm1.xml_tree_to_prefix_string(F);
1311: basic_plus(basic_times(basic_power(x,4),1),basic_times(basic_power(x,0),-1))
1312: @end example
1313: */
1314: /*&en
1315: @table @t
1316: @item Reference
1317: @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
1318: @end table
1319: */
1320: /*&ja
1321: @table @t
1322: @item $B;2>H(B
1323: @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
1324: @end table
1325: */
1326:
1327:
1328:
1329:
1330: /*&en
1331: @c sort-sm1.syz
1332: @node sm1.syz,,, SM1 Functions
1333: @subsection @code{sm1.syz}
1334: @findex sm1.syz
1335: @findex sm1.syz_d
1336: @table @t
1337: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1338: :: computes the syzygy of @var{f} in the ring of differential
1339: operators with the variable @var{v}.
1340: @end table
1341:
1342: @table @var
1343: @item return
1344: List
1345: @item p
1346: Number
1347: @item f, v, w
1348: List
1349: @end table
1350:
1351: @itemize @bullet
1352: @item
1353: The return values is of the form
1354: [@var{s},[@var{g}, @var{m}, @var{t}]].
1355: Here @var{s} is the syzygy of @var{f} in the ring of differential
1356: operators with the variable @var{v}.
1357: @var{g} is a Groebner basis of @var{f} with the weight vector @var{w},
1358: and @var{m} is a matrix that translates the input matrix @var{f} to the Gr\"obner
1359: basis @var{g}.
1360: @var{t} is the syzygy of the Gr\"obner basis @var{g}.
1361: In summary, @var{g} = @var{m} @var{f} and
1362: @var{s} @var{f} = 0 hold as matrices.
1363: @item
1364: The weight vectors are given by @var{w}, which can be omitted.
1365: If @var{w} is not given,
1366: the graded reverse lexicographic order will be used to compute Grobner basis.
1367: @item
1368: When a non-term order is given, the Grobner basis is computed in
1369: the homogenized Weyl algebra (See Section 1.2 of the book of SST).
1370: The homogenization variable h is automatically added.
1371: @end itemize
1372: */
1373: /*&ja
1374: @c sort-sm1.syz
1375: @node sm1.syz,,, SM1 Functions
1376: @subsection @code{sm1.syz}
1377: @findex sm1.syz
1378: @findex sm1.syz_d
1379: @table @t
1380: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1381: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N(B syzygy $B$r7W;;$9$k(B.
1382: @end table
1383:
1384: @table @var
1385: @item return
1386: $B%j%9%H(B
1387: @item p
1388: $B?t(B
1389: @item f, v, w
1390: $B%j%9%H(B
1391: @end table
1392:
1393: @itemize @bullet
1394: @item
1395: $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
1396: [@var{s},[@var{g}, @var{m}, @var{t}]].
1397: $B$3$3$G(B @var{s} $B$O(B @var{f} $B$N(B @var{v} $B$rJQ?t$H$9$kHyJ,:nMQAG4D$K$*$1$k(B
1398: syzygy $B$G$"$k(B.
1399: @var{g} $B$O(B @var{f} $B$N(B weight vector @var{w} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
1400: @var{m} $B$OF~NO9TNs(B @var{f} $B$r%0%l%V%J4pDl(B
1401: @var{g} $B$XJQ49$9$k9TNs$G$"$k(B.
1402: @var{t} $B$O%0%l%V%J4pDl(B @var{g} $B$N(B syzygy $B$G$"$k(B.
1403: $B$^$H$a$k$H(B, $B<!$NEy<0$,$J$j$?$D(B:
1404: @var{g} = @var{m} @var{f} ,
1405: @var{s} @var{f} = 0.
1406: @item
1407: Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
1408: $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
1409: $B%V%l%V%J4pDl$r7W;;$9$k(B.
1410: @item
1411: 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).
1412: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
1413: @end itemize
1414: */
1415: /*&C
1416: @example
1417: [293] sm1.syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
1418: [[[y*x*dy*dx-2,-x*dx-y*dy+1]], generators of the syzygy
1419: [[[x*dx+y*dy-1],[y^2*dy^2+2]], grobner basis
1420: [[1,0],[y*dy,-1]], transformation matrix
1421: [[y*x*dy*dx-2,-x*dx-y*dy+1]]]]
1422: @end example
1423: */
1424: /*&C
1425: @example
1426: [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]]]);
1427: [[[y*x*dy*dx-1,-x^2*dx^2-x*dx-y^2*dy^2-y*dy+4]], generators of the syzygy
1428: [[[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
1429: [h^4*x*dx+y^3*dy^3+3*h^2*y^2*dy^2-3*h^4*y*dy]],
1430: [[1,0],[0,1],[y*dy,-x*dx]], transformation matrix
1431: [[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]]]]
1432: @end example
1433: */
1434:
1435:
1436:
1437: /*&en
1438: @node sm1.mul,,, SM1 Functions
1439: @subsection @code{sm1.mul}
1440: @findex sm1.mul
1441: @table @t
1442: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1443: :: ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
1444: @end table
1445:
1446: @table @var
1447: @item return
1448: Polynomial or List
1449: @item p
1450: Number
1451: @item f, g
1452: Polynomial or List
1453: @item v
1454: List
1455: @end table
1456:
1457: @itemize @bullet
1458: @item Ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
1459: @item @code{sm1.mul_h} is for homogenized Weyl algebra.
1460: @item BUG: @code{sm1.mul(p0*dp0,1,[p0])} returns
1461: @code{dp0*p0+1}.
1462: A variable order such that d-variables come after non-d-variables
1463: is necessary for the correct computation.
1464: @end itemize
1465: */
1466:
1467: /*&ja
1468: @node sm1.mul,,, SM1 Functions
1469: @subsection @code{sm1.mul}
1470: @findex sm1.mul
1471: @table @t
1472: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1473: :: sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
1474: $B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
1475: @end table
1476:
1477: @table @var
1478: @item return
1479: $BB?9`<0$^$?$O%j%9%H(B
1480: @item p
1481: $B?t(B
1482: @item f, g
1483: $BB?9`<0$^$?$O%j%9%H(B
1484: @item v
1485: $B%j%9%H(B
1486: @end table
1487:
1488: @itemize @bullet
1489: @item sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
1490: $B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
1491: @item @code{sm1.mul_h} $B$O(B homogenized Weyl $BBe?tMQ(B.
1492: @item BUG: @code{sm1.mul(p0*dp0,1,[p0])} $B$O(B
1493: @code{dp0*p0+1} $B$rLa$9(B.
1494: 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.
1495: @end itemize
1496: */
1497:
1498: /*&C
1499:
1500: @example
1501: [277] sm1.mul(dx,x,[x]);
1502: x*dx+1
1503: [278] sm1.mul([x,y],[1,2],[x,y]);
1504: x+2*y
1505: [279] sm1.mul([[1,2],[3,4]],[[x,y],[1,2]],[x,y]);
1506: [[x+2,y+4],[3*x+4,3*y+8]]
1507: @end example
1508:
1509: */
1510:
1511:
1512:
1513:
1514: /*&en
1515: @node sm1.distraction,,, SM1 Functions
1516: @subsection @code{sm1.distraction}
1517: @findex sm1.distraction
1518: @table @t
1519: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1520: :: ask the @code{sm1} server to compute the distraction of @var{f}.
1521: @end table
1522:
1523: @table @var
1524: @item return
1525: List
1526: @item p
1527: Number
1528: @item f
1529: Polynomial
1530: @item v,x,d,s
1531: List
1532: @end table
1533:
1534: @itemize @bullet
1535: @item It asks the @code{sm1} server of the descriptor number @var{p}
1536: to compute the distraction of @var{f} in the ring of differential
1537: operators with variables @var{v}.
1538: @item @var{x} is a list of x-variables and @var{d} is that of d-variables
1539: to be distracted. @var{s} is a list of variables to express the distracted @var{f}.
1540: @item Distraction is roughly speaking to replace x*dx by a single variable x.
1541: See Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations at page 68 for details.
1542: @end itemize
1543: */
1544:
1545: /*&ja
1546: @node sm1.distraction,,, SM1 Functions
1547:
1548: @subsection @code{sm1.distraction}
1549: @findex sm1.distraction
1550: @table @t
1551: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1552: :: @code{sm1} $B$K(B @var{f} $B$N(B distraction $B$r7W;;$7$F$b$i$&(B.
1553: @end table
1554:
1555: @table @var
1556: @item return
1557: $B%j%9%H(B
1558: @item p
1559: $B?t(B
1560: @item f
1561: $BB?9`<0(B
1562: @item v,x,d,s
1563: $B%j%9%H(B
1564: @end table
1565:
1566: @itemize @bullet
1567: @item $B<1JL;R(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B,
1568: @var{f} $B$N(B distraction $B$r(B @var{v} $B>e$NHyJ,:nMQAG4D$G7W;;$7$F$b$i$&(B.
1569: @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
1570: $B%j%9%H(B. Distraction $B$7$?$i(B, @var{s} $B$rJQ?t$H$7$F7k2L$rI=$9(B.
1571: @item Distraction $B$H$$$&$N$O(B x*dx $B$r(B x $B$GCV$-49$($k$3$H$G$"$k(B.
1572: $B>\$7$/$O(B Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations $B$N(B page 68 $B$r8+$h(B.
1573: @end itemize
1574: */
1575:
1576: /*&C
1577:
1578: @example
1579: [280] sm1.distraction([x*dx,[x],[x],[dx],[x]]);
1580: x
1581: [281] sm1.distraction([dx^2,[x],[x],[dx],[x]]);
1582: x^2-x
1583: [282] sm1.distraction([x^2,[x],[x],[dx],[x]]);
1584: x^2+3*x+2
1585: [283] fctr(@@);
1586: [[1,1],[x+1,1],[x+2,1]]
1587: [284] sm1.distraction([x*dx*y+x^2*dx^2*dy,[x,y],[x],[dx],[x]]);
1588: (x^2-x)*dy+x*y
1589: @end example
1590: */
1591:
1592: /*&en
1593: @table @t
1594: @item Reference
1595: @code{distraction2(sm1)},
1596: @end table
1597: */
1598:
1599: /*&ja
1600: @table @t
1601: @item $B;2>H(B
1602: @code{distraction2(sm1)},
1603: @end table
1604: */
1605:
1606:
1607:
1608: /*&en
1609: @node sm1.gkz,,, SM1 Functions
1610: @subsection @code{sm1.gkz}
1611: @findex sm1.gkz
1612: @table @t
1613: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1614: :: Returns the GKZ system (A-hypergeometric system) associated to the matrix
1615: @var{A} with the parameter vector @var{B}.
1616: @end table
1617:
1618: @table @var
1619: @item return
1620: List
1621: @item p
1622: Number
1623: @item A, B
1624: List
1625: @end table
1626:
1627: @itemize @bullet
1628: @item Returns the GKZ hypergeometric system
1629: (A-hypergeometric system) associated to the matrix
1630: @end itemize
1631: */
1632:
1633: /*&ja
1634: @node sm1.gkz,,, SM1 Functions
1635: @subsection @code{sm1.gkz}
1636: @findex sm1.gkz
1637: @table @t
1638: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1639: :: $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.
1640: @end table
1641:
1642: @table @var
1643: @item return
1644: $B%j%9%H(B
1645: @item p
1646: $B?t(B
1647: @item A, B
1648: $B%j%9%H(B
1649: @end table
1650:
1651: @itemize @bullet
1652: @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.
1653: @end itemize
1654: */
1655:
1656: /*&C
1657:
1658: @example
1659:
1660: [280] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
1661: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1662: -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1663: [x1,x2,x3,x4]]
1664:
1665: @end example
1666:
1667: */
1668:
1.2 takayama 1669: /*&en
1670: @node sm1.mgkz,,, SM1 Functions
1671: @subsection @code{sm1.mgkz}
1672: @findex sm1.mgkz
1673: @table @t
1674: @item sm1.mgkz([@var{A},@var{W},@var{B}]|proc=@var{p})
1675: :: Returns the modified GKZ system (A-hypergeometric system) associated to the matrix
1676: @var{A} and the weight @var{w} with the parameter vector @var{B}.
1677: @end table
1678:
1679: @table @var
1680: @item return
1681: List
1682: @item p
1683: Number
1684: @item A, W, B
1685: List
1686: @end table
1687:
1688: @itemize @bullet
1689: @item Returns the modified GKZ hypergeometric system
1690: (A-hypergeometric system) associated to the matrix
1691: @item http://arxiv.org/abs/0707.0043
1692: @end itemize
1693: */
1694:
1695: /*&ja
1696: @node sm1.mgkz,,, SM1 Functions
1697: @subsection @code{sm1.mgkz}
1698: @findex sm1.mgkz
1699: @table @t
1700: @item sm1.mgkz([@var{A},@var{W},@var{B}]|proc=@var{p})
1701: :: $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.
1702: @end table
1703:
1704: @table @var
1705: @item return
1706: $B%j%9%H(B
1707: @item p
1708: $B?t(B
1709: @item A, W, B
1710: $B%j%9%H(B
1711: @end table
1712:
1713: @itemize @bullet
1714: @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.
1715: @item http://arxiv.org/abs/0707.0043
1716: @end itemize
1717: */
1718:
1719: /*&C
1720:
1721: @example
1722:
1723: [280] sm1.mgkz([ [[1,2,3]], [1,2,1], [a/2]]);
1724: [[6*x3*dx3+4*x2*dx2+2*x1*dx1-a,-x4*dx4+x3*dx3+2*x2*dx2+x1*dx1,
1725: -dx2+dx1^2,-x4^2*dx3+dx1*dx2],[x1,x2,x3,x4]]
1726:
1727: Modified A-hypergeometric system for
1728: A=(1,2,3), w=(1,2,1), beta=(a/2).
1729: @end example
1730:
1731: */
1732:
1.1 takayama 1733:
1734:
1735:
1736: /*&en
1737: @node sm1.appell1,,, SM1 Functions
1738: @subsection @code{sm1.appell1}
1739: @findex sm1.appell1
1740: @table @t
1741: @item sm1.appell1(@var{a}|proc=@var{p})
1742: :: Returns the Appell hypergeometric system F_1 or F_D.
1743: @end table
1744:
1745: @table @var
1746: @item return
1747: List
1748: @item p
1749: Number
1750: @item a
1751: List
1752: @end table
1753:
1754: @itemize @bullet
1755: @item Returns the hypergeometric system for the Lauricella function
1756: F_D(a,b1,b2,...,bn,c;x1,...,xn)
1757: where @var{a} =(a,c,b1,...,bn).
1758: When n=2, the Lauricella function is called the Appell function F_1.
1759: The parameters a, c, b1, ..., bn may be rational numbers.
1760: @item It does not call sm1 function appell1. As a concequence,
1761: when parameters are rational or symbolic, this function also works
1762: as well as integral parameters.
1763: @end itemize
1764: */
1765:
1766: /*&ja
1767: @node sm1.appell1,,, SM1 Functions
1768: @subsection @code{sm1.appell1}
1769: @findex sm1.appell1
1770: @table @t
1771: @item sm1.appell1(@var{a}|proc=@var{p})
1772: :: F_1 $B$^$?$O(B F_D $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
1773: @end table
1774:
1775: @table @var
1776: @item return
1777: $B%j%9%H(B
1778: @item p
1779: $B?t(B
1780: @item a
1781: $B%j%9%H(B
1782: @end table
1783:
1784: @itemize @bullet
1785: @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
1786: F_D(a,b1,b2,...,bn,c;x1,...,xn)
1787: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
1788: @var{a} =(a,c,b1,...,bn).
1789: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
1790: @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
1791: $B@5$7$/F0$/(B.
1792: @end itemize
1793: */
1794:
1795: /*&C
1796:
1797: @example
1798:
1799: [281] sm1.appell1([1,2,3,4]);
1800: [[((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2+(-5*x1+2)*dx1-3,
1801: (-x2^2+x2)*dx2^2+((-x1*x2+x1)*dx1-6*x2+2)*dx2-4*x1*dx1-4,
1802: ((-x2+x1)*dx1+3)*dx2-4*dx1], equations
1803: [x1,x2]] the list of variables
1804:
1805: [282] sm1.gb(@@);
1806: [[((-x2+x1)*dx1+3)*dx2-4*dx1,((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2
1807: +(-5*x1+2)*dx1-3,(-x2^2+x2)*dx2^2+((-x2^2+x1)*dx1-3*x2+2)*dx2
1808: +(-4*x2-4*x1)*dx1-4,
1809: (x2^3+(-x1-1)*x2^2+x1*x2)*dx2^2+((-x1*x2+x1^2)*dx1+6*x2^2
1810: +(-3*x1-2)*x2+2*x1)*dx2-4*x1^2*dx1+4*x2-4*x1],
1811: [x1*dx1*dx2,-x1^2*dx1^2,-x2^2*dx1*dx2,-x1*x2^2*dx2^2]]
1812:
1813: [283] sm1.rank(sm1.appell1([1/2,3,5,-1/3]));
1814: 3
1815:
1816: [285] Mu=2$ Beta = 1/3$
1817: [287] sm1.rank(sm1.appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));
1818: 4
1819:
1820:
1821: @end example
1822:
1823: */
1824:
1825: /*&en
1826: @node sm1.appell4,,, SM1 Functions
1827: @subsection @code{sm1.appell4}
1828: @findex sm1.appell4
1829: @table @t
1830: @item sm1.appell4(@var{a}|proc=@var{p})
1831: :: Returns the Appell hypergeometric system F_4 or F_C.
1832: @end table
1833:
1834: @table @var
1835: @item return
1836: List
1837: @item p
1838: Number
1839: @item a
1840: List
1841: @end table
1842:
1843: @itemize @bullet
1844: @item Returns the hypergeometric system for the Lauricella function
1845: F_4(a,b,c1,c2,...,cn;x1,...,xn)
1846: where @var{a} =(a,b,c1,...,cn).
1847: When n=2, the Lauricella function is called the Appell function F_4.
1848: The parameters a, b, c1, ..., cn may be rational numbers.
1849: @item @item It does not call sm1 function appell4. As a concequence,
1850: when parameters are rational or symbolic, this function also works
1851: as well as integral parameters.
1852: @end itemize
1853: */
1854:
1855: /*&ja
1856: @node sm1.appell4,,, SM1 Functions
1857: @subsection @code{sm1.appell4}
1858: @findex sm1.appell4
1859: @table @t
1860: @item sm1.appell4(@var{a}|proc=@var{p})
1861: :: F_4 $B$^$?$O(B F_C $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
1862: @end table
1863:
1864: @table @var
1865: @item return
1866: $B%j%9%H(B
1867: @item p
1868: $B?t(B
1869: @item a
1870: $B%j%9%H(B
1871: @end table
1872:
1873: @itemize @bullet
1874: @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
1875: F_C(a,b,c1,c2,...,cn;x1,...,xn)
1876: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
1877: @var{a} =(a,b,c1,...,cn).
1878: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
1879: @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
1880: $B@5$7$/F0$/(B.
1881: @end itemize
1882: */
1883:
1884: /*&C
1885:
1886: @example
1887:
1888: [281] sm1.appell4([1,2,3,4]);
1889: [[-x2^2*dx2^2+(-2*x1*x2*dx1-4*x2)*dx2+(-x1^2+x1)*dx1^2+(-4*x1+3)*dx1-2,
1890: (-x2^2+x2)*dx2^2+(-2*x1*x2*dx1-4*x2+4)*dx2-x1^2*dx1^2-4*x1*dx1-2],
1891: equations
1892: [x1,x2]] the list of variables
1893:
1894: [282] sm1.rank(@@);
1895: 4
1896:
1897: @end example
1898:
1899: */
1900:
1901:
1902:
1903:
1904: /*&en
1905: @node sm1.rank,,, SM1 Functions
1906: @subsection @code{sm1.rank}
1907: @findex sm1.rank
1908: @table @t
1909: @item sm1.rank(@var{a}|proc=@var{p})
1910: :: Returns the holonomic rank of the system of differential equations @var{a}.
1911: @end table
1912:
1913: @table @var
1914: @item return
1915: Number
1916: @item p
1917: Number
1918: @item a
1919: List
1920: @end table
1921:
1922: @itemize @bullet
1923: @item It evaluates the dimension of the space of holomorphic solutions
1924: at a generic point of the system of differential equations @var{a}.
1925: The dimension is called the holonomic rank.
1926: @item @var{a} is a list consisting of a list of differential equations and
1927: a list of variables.
1928: @item @code{sm1.rrank} returns the holonomic rank when @var{a} is regular
1929: holonomic. It is generally faster than @code{sm1.rank}.
1930: @end itemize
1931: */
1932:
1933: /*&ja
1934: @node sm1.rank,,, SM1 Functions
1935: @subsection @code{sm1.rank}
1936: @findex sm1.rank
1937: @table @t
1938: @item sm1.rank(@var{a}|proc=@var{p})
1939: :: $BHyJ,J}Dx<07O(B @var{a} $B$N(B holonomic rank $B$rLa$9(B.
1940: @end table
1941:
1942: @table @var
1943: @item return
1944: $B?t(B
1945: @item p
1946: $B?t(B
1947: @item a
1948: $B%j%9%H(B
1949: @end table
1950:
1951: @itemize @bullet
1952: @item $BHyJ,J}Dx<07O(B @var{a} $B$N(B, generic point $B$G$N@5B'2r$N<!85$r(B
1953: $BLa$9(B. $B$3$N<!85$r(B, holonomic rank $B$H8F$V(B.
1954: @item @var{a} $B$OHyJ,:nMQAG$N%j%9%H$HJQ?t$N%j%9%H$h$j$J$k(B.
1955: @item @var{a} $B$,(B regular holonomic $B$N$H$-$O(B @code{sm1.rrank}
1956: $B$b(B holonomic rank $B$rLa$9(B.
1957: $B$$$C$Q$s$K$3$N4X?t$NJ}$,(B @code{sm1.rank} $B$h$jAa$$(B.
1958: @end itemize
1959: */
1960:
1961: /*&C
1962:
1963: @example
1964:
1965: [284] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
1966: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1967: -dx1*dx4+dx2*dx3, -dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1968: [x1,x2,x3,x4]]
1969: [285] sm1.rrank(@@);
1970: 4
1971:
1972: [286] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [1,2]]);
1973: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1974: -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1975: [x1,x2,x3,x4]]
1976: [287] sm1.rrank(@@);
1977: 5
1978:
1979: @end example
1980:
1981: */
1982:
1983:
1984: /*&en
1985: @node sm1.auto_reduce,,, SM1 Functions
1986: @subsection @code{sm1.auto_reduce}
1987: @findex sm1.auto_reduce
1988: @table @t
1989: @item sm1.auto_reduce(@var{s}|proc=@var{p})
1990: :: Set the flag "AutoReduce" to @var{s}.
1991: @end table
1992:
1993: @table @var
1994: @item return
1995: Number
1996: @item p
1997: Number
1998: @item s
1999: Number
2000: @end table
2001:
2002: @itemize @bullet
2003: @item If @var{s} is 1, then all Grobner bases to be computed
2004: will be the reduced Grobner bases.
2005: @item If @var{s} is 0, then Grobner bases are not necessarily the reduced
2006: Grobner bases. This is the default.
2007: @end itemize
2008: */
2009:
2010: /*&ja
2011: @node sm1.auto_reduce,,, SM1 Functions
2012: @subsection @code{sm1.auto_reduce}
2013: @findex sm1.auto_reduce
2014: @table @t
2015: @item sm1.auto_reduce(@var{s}|proc=@var{p})
2016: :: $B%U%i%0(B "AutoReduce" $B$r(B @var{s} $B$K@_Dj(B.
2017: @end table
2018:
2019: @table @var
2020: @item $BLa$jCM(B
2021: $B?t(B
2022: @item p
2023: $B?t(B
2024: @item s
2025: $B?t(B
2026: @end table
2027:
2028: @itemize @bullet
2029: @item @var{s} $B$,(B 1 $B$N$H$-(B, $B0J8e7W;;$5$l$k%0%l%V%J4pDl$O$9$Y$F(B,
2030: reduced $B%0%l%V%J4pDl$H$J$k(B.
2031: @item @var{s} $B$,(B 0 $B$N$H$-(B, $B7W;;$5$l$k%0%l%V%J4pDl$O(B
2032: reduced $B%0%l%V%J4pDl$H$O$+$.$i$J$$(B. $B$3$A$i$,%G%U%)!<%k%H(B.
2033: @end itemize
2034: */
2035:
2036:
2037:
2038: /*&en
2039: @node sm1.slope,,, SM1 Functions
2040: @subsection @code{sm1.slope}
2041: @findex sm1.slope
2042: @table @t
2043: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
2044: :: Returns the slopes of differential equations @var{ii}.
2045: @end table
2046:
2047: @table @var
2048: @item return
2049: List
2050: @item p
2051: Number
2052: @item ii
2053: List (equations)
2054: @item v
2055: List (variables)
2056: @item f_filtration
2057: List (weight vector)
2058: @item v_filtration
2059: List (weight vector)
2060: @end table
2061:
2062: @itemize @bullet
2063: @item @code{sm1.slope} returns the (geometric) slopes
2064: of the system of differential equations @var{ii}
2065: along the hyperplane specified by
2066: the V filtration @var{v_filtration}.
2067: @item @var{v} is a list of variables.
2068: @item The return value is a list of lists.
2069: The first entry of each list is the slope and the second entry
2070: is the weight vector for which the microcharacteristic variety is
2071: not bihomogeneous.
2072: @end itemize
2073:
2074: @noindent
2075: Algorithm:
2076: see "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
2077: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
1.4 ! takayama 2078: Note that the signs of the slopes s' are negative, but the absolute values -s'
1.1 takayama 2079: of the slopes are returned.
1.4 ! takayama 2080: In other words, when pF+qV is the gap, -s'=q/p is returned.
! 2081: Note that s=1-1/s' is called the slope in recent literatures. Solutions belongs to O(s).
! 2082: The number s satisfies 1<= s.
! 2083: 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)
! 2084: in the Borel and Laplace transformations respectively.
1.1 takayama 2085:
2086: */
2087:
2088: /*&ja
2089: @node sm1.slope,,, SM1 Functions
2090: @subsection @code{sm1.slope}
2091: @findex sm1.slope
2092: @table @t
2093: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
2094: :: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B slope $B$rLa$9(B.
2095: @end table
2096:
2097: @table @var
2098: @item return
2099: $B?t(B
2100: @item p
2101: $B?t(B
2102: @item ii
2103: $B%j%9%H(B ($BJ}Dx<0(B)
2104: @item v
2105: $B%j%9%H(B ($BJQ?t(B)
2106: @item f_filtration
2107: $B%j%9%H(B (weight vector)
2108: @item v_filtration
2109: $B%j%9%H(B (weight vector)
2110: @end table
2111:
2112: @itemize @bullet
2113: @item @code{sm1.slope} $B$O(B
2114: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B V filtration @var{v_filtration}
2115: $B$G;XDj$9$kD6J?LL$K1h$C$F$N(B (geomeric) slope $B$r7W;;$9$k(B.
2116: @item @var{v} $B$OJQ?t$N%j%9%H(B.
2117: @item $BLa$jCM$O(B, $B%j%9%H$r@.J,$H$9$k%j%9%H$G$"$k(B.
2118: $B@.J,%j%9%H$NBh(B 1 $BMWAG$,(B slope, $BBh(B 2 $BMWAG$O(B, $B$=$N(B weight vector $B$KBP1~$9$k(B
2119: microcharacteristic variety $B$,(B bihomogeneous $B$G$J$$(B.
2120: @end itemize
2121:
2122: @noindent
2123: Algorithm:
2124: "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
2125: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
2126: $B$r$_$h(B.
1.4 ! takayama 2127: Slope s' $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B, $B$3$N%W%m%0%i%`$O(B,
! 2128: Slope $B$N@dBPCM(B -s' $B$rLa$9(B.
! 2129: $B$D$^$j(B pF+qV $B$,(Bmicro$BFC@-B?MMBN$N(Bgap$B$G$"$k$H$-(B, -s'=q/p $B$rLa$9(B.
! 2130: $B:G6a$NJ88%$G$O(B s=1-1/s' $B$r(B slope $B$H8F$s$G$$$k(B. $B2r$O(B O(s) $B$KB0$9$k(B.
! 2131: $B?t(B s $B$O(B 1<= s $B$rK~$9(B.
! 2132: r=s-1=-1/s' $B$*$h$S(B kappa=1/r=-s' $B$G$"$k(B.
! 2133: $B$3$l$i$N?t$O(BBorel and Laplace $BJQ49$K$*$$$F$=$l$>$l(B 1/Gamma(1+m*r) factor,
! 2134: exp(-tau^kappa) $B9`$H$7$F;H$o$l$k(B.
1.1 takayama 2135: */
2136:
2137: /*&C
2138:
2139: @example
2140:
2141: [284] A= sm1.gkz([ [[1,2,3]], [-3] ]);
2142:
2143:
2144: [285] sm1.slope(A[0],A[1],[0,0,0,1,1,1],[0,0,-1,0,0,1]);
2145:
2146: [286] A2 = sm1.gkz([ [[1,1,1,0],[2,-3,1,-3]], [1,0]]);
2147: (* This is an interesting example given by Laura Matusevich,
2148: June 9, 2001 *)
2149:
2150: [287] sm1.slope(A2[0],A2[1],[0,0,0,0,1,1,1,1],[0,0,0,-1,0,0,0,1]);
2151:
2152:
2153: @end example
2154:
2155: */
2156: /*&en
2157: @table @t
2158: @item Reference
2159: @code{sm.gb}
2160: @end table
2161: */
2162: /*&ja
2163: @table @t
2164: @item $B;2>H(B
2165: @code{sm.gb}
2166: @end table
2167: */
2168:
2169:
2170: /*&en
2171: @include sm1-auto.en
2172: */
2173:
2174: /*&ja
2175: @include sm1-auto.ja
2176: */
2177:
2178:
2179: end$
2180:
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