Annotation of OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw, Revision 1.5
1.5 ! takayama 1: /*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1/sm1.oxw,v 1.4 2012/06/11 05:23:52 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).
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.
1.1 takayama 564: @end itemize
565: */
566: /*&ja
567: @c sort-sm1.gb
568: @node sm1.gb,,, SM1 Functions
569: @subsection @code{sm1.gb}
570: @findex sm1.gb
571: @findex sm1.gb_d
572: @table @t
573: @item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
574: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
575: @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
576: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$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.
577: @end table
578:
579: @table @var
580: @item return
581: $B%j%9%H(B
582: @item p, q, r
583: $B?t(B
584: @item f, v, w
585: $B%j%9%H(B
586: @end table
587:
588: @itemize @bullet
589: @item
590: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
591: @item
592: Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
593: $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
594: $B%V%l%V%J4pDl$r7W;;$9$k(B.
595: @item
596: @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
597: ( @var{w} $B$,$J$$$H$-(B ) $B$^$?$O(B $B%$%K%7%!%kB?9`<0(B ( @var{w} $B$,M?$($i$?$H$-(B)
598: $B$N%j%9%H$G$"$k(B.
599: @item
600: @code{sm1.gb_d} $B$O7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.
601: $BB?9`<0$NCf$K8=$l$k%b%N%_%"%k$O%0%l%V%J4pDl$r7W;;$9$k$H$-$KM?$($i$?=g=x$G%=!<%H$5$l$F$$$k(B.
602: $BLa$jCM$O(B
603: [$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]
604: $B$G$"$k(B.
605: @item
606: Term order $B$G$J$$=g=x$,M?$($i$l$?>l9g$O(B, $BF1<!2=%o%$%kBe?t$G%0%l%V%J4pDl$,7W;;$5$l$k(B (SST $B$NK\$N(B Section 1.2 $B$r8+$h(B).
607: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
608: @item $B%*%W%7%g%J%kJQ?t(B @var{q} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
609: 3 $BHVL\$NLa$jCM$H$7$F(B, $B%0%l%V%J4pDl$*$h$S%$%K%7%!%k$N%j%9%H$,(B
610: $BM?$($i$l$?=g=x$G%=!<%H$5$l$?%b%N%_%"%k$NOB$H$7$FLa$5$l$k(B.
611: $B$$$^$N$H$3$m$3$NB?9`<0$O(B, $BJ8;zNs$GI=8=$5$l$k(B.
612: $B%*%W%7%g%J%kJQ?t(B @var{r} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
613: $BLa$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 614: @item Reduced $B%0%l%V%J!<4pDl$^$?$O(B in_w $B$r7W;;$7$?$$$H$-$O(B, $B$3$N4X?t$N<B9T$NA0$K(B
! 615: sm1.auto_reduce(1) $B$r<B9T$7$F$*$/$3$H(B.
1.1 takayama 616: @end itemize
617: */
618: /*&C
619: @example
620: [293] sm1.gb([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
621: [[x*dx+y*dy-1,y^2*dy^2+2],[x*dx,y^2*dy^2]]
622: @end example
623: */
624: /*&en
625: In the example above,
626: @tex the set $\{ x \partial_x + y \partial_y -1,
627: y^2 \partial_y^2+2\}$
628: is the Gr\"obner basis of the input with respect to the
629: graded reverse lexicographic order such that
630: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$.
631: The set $\{x \partial_x, y^2 \partial_y\}$ is the leading monomials
632: (the initial monominals) of the Gr\"obner basis.
633: @end tex
634: */
635: /*&ja
636: $B>e$NNc$K$*$$$F(B,
637: @tex $B=89g(B $\{ x \partial_x + y \partial_y -1,
638: y^2 \partial_y^2+2\}$
639: $B$O(B
640: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$
641: $B$G$"$k$h$&$J(B
642: graded reverse lexicographic order $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
643: $B=89g(B $\{x \partial_x, y^2 \partial_y\}$ $B$O%0%l%V%J4pDl$N3F85$K(B
644: $BBP$9$k(B leading monomial (initial monomial) $B$G$"$k(B.
645: @end tex
646: */
647: /*&C
648: @example
649: [294] sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]);
650: [[dx+dy^3-4*dy,-dy^4+4*dy^2-1],[dx,-dy^4]]
651: @end example
652: */
653: /*&en
654: In the example above, two monomials
655: @tex
656: $m = x^a y^b \partial_x^c \partial_y^d$ and
657: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
658: are firstly compared by the weight vector
659: {\tt (dx,dy,x,y) = (50,2,1,0)}
660: (i.e., $m$ is larger than $m'$ if $50c+2d+a > 50c'+2d'+a'$)
661: and when the comparison is tie, then these are
662: compared by the reverse lexicographic order
663: (i.e., if $50c+2d+a = 50c'+2d'+a'$, then use the reverse lexicogrpahic order).
664: @end tex
665: */
666: /*&ja
667: $B>e$NNc$K$*$$$FFs$D$N%b%N%_%"%k(B
668: @tex
669: $m = x^a y^b \partial_x^c \partial_y^d$ $B$*$h$S(B
670: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
671: $B$O:G=i$K(B weight vector
672: {\tt (dx,dy,x,y) = (50,2,1,0)} $B$rMQ$$$FHf3S$5$l$k(B
673: ($B$D$^$j(B $m$ $B$O(B $50c+2d+a > 50c'+2d'+a'$ $B$N$H$-(B
674: $m'$ $B$h$jBg$-$$(B )
675: $B<!$K$3$NHf3S$G>!Ii$,$D$+$J$$$H$-$O(B reverse lexicographic order $B$GHf3S$5$l$k(B
676: ($B$D$^$j(B $50c+2d+a = 50c'+2d'+a'$ $B$N$H$-(B reverse lexicographic order $B$GHf3S(B
677: $B$5$l$k(B).
678: @end tex
679: */
680: /*&C
681: @example
682: [294] F=sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);
683: map(print,F[2][0])$
684: map(print,F[2][1])$
685: @end example
686: */
687: /*&C
688: @example
689: [595]
690: sm1.gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],
691: [x,y],[[dx,1,x,-1],[dy,1]]]);
692:
693: [[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],
694: [x*dx^2+(y*dy-h^2)*dx,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx]]
695:
696: [596]
697: sm1.gb_d([["dx (x dx +y dy-2)-1","dy (x dx + y dy -2)-1"],
698: "x,y",[[dx,1,x,-1],[dy,1]]]);
699: [[[e0,x,y,H,E,dx,dy,h],
700: [[0,-1,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],
701: [0,1,1,1,1,1,1,0],[0,0,0,0,0,0,-1,0],[0,0,0,0,0,-1,0,0],
702: [0,0,0,0,-1,0,0,0],[0,0,0,-1,0,0,0,0],[0,0,-1,0,0,0,0,0],
703: [0,0,0,0,0,0,0,1]]],
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: <<0,0,0,0,0,0,0,3>>,(1)*<<0,0,1,0,0,0,2,0>>+(1)*<<0,1,0,0,0,1,1,0>>+(-1)*<<0,0,0
706: ,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
707: ,1,3>>],
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: <0,0,1,0,0,0,2,0>>+(1)*<<0,1,0,0,0,1,1,0>>+(-1)*<<0,0,0,0,0,0,1,2>>+(-1)*<<0,0,0
710: ,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]]
711: @end example
712: */
713:
714: /*&en
715: @table @t
716: @item Reference
1.5 ! takayama 717: @code{sm1.auto_reduce}, @code{sm1.reduction}, @code{sm1.rat_to_p}
1.1 takayama 718: @end table
719: */
720: /*&ja
721: @table @t
722: @item $B;2>H(B
1.5 ! takayama 723: @code{sm1.auto_reduce}, @code{sm1.reduction}, @code{sm1.rat_to_p}
1.1 takayama 724: @end table
725: */
726:
727:
728:
729: /*&en
730: @c sort-sm1.deRham
731: @node sm1.deRham,,, SM1 Functions
732: @subsection @code{sm1.deRham}
733: @findex sm1.deRham
734: @table @t
735: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
736: :: ask the server to evaluate the dimensions of the de Rham cohomology groups
737: of C^n - (the zero set of @var{f}=0).
738: @end table
739:
740: @table @var
741: @item return
742: List
743: @item p
744: Number
745: @item f
746: String or polynomial
747: @item v
748: List
749: @end table
750:
751: @itemize @bullet
752: @item It returns the dimensions of the de Rham cohomology groups
753: of X = C^n \ V(@var{f}).
754: In other words, it returns
755: [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)].
756: @item @var{v} is a list of variables. n = @code{length(@var{v})}.
757: @item
758: @code{sm1.deRham} requires huge computer resources.
759: For example, @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
760: is already very hard.
761: @item
762: To efficiently analyze the roots of b-function, @code{ox_asir} should be used
763: from @code{ox_sm1_forAsir}.
764: It is recommended to load the communication module for @code{ox_asir}
765: by the command @*
766: @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
767: This command is automatically executed when @code{ox_sm1_forAsir} is started.
768: @item If you make an interruption to the function @code{sm1.deRham}
769: by @code{ox_reset(sm1.get_Sm1_proc());}, the server might get out of the standard
770: mode. So, it is strongly recommended to execute the command
771: @code{ox_shutdown(sm1.get_Sm1_proc());} to interrupt and restart the server.
772: @end itemize
773: */
774: /*&ja
775: @c sort-sm1.deRham
776: @node sm1.deRham,,, SM1 Functions
777: @subsection @code{sm1.deRham}
778: @findex sm1.deRham
779: @table @t
780: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
781: :: $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.
782: @end table
783:
784: @table @var
785: @item return
786: $B%j%9%H(B
787: @item p
788: $B?t(B
789: @item f
790: $BJ8;zNs(B $B$^$?$O(B $BB?9`<0(B
791: @item v
792: $B%j%9%H(B
793: @end table
794:
795: @itemize @bullet
796: @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.
797: $B$9$J$o$A(B,
798: [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)]
799: $B$rLa$9(B.
800: @item @var{v} $B$OJQ?t$N%j%9%H(B. n = @code{length(@var{v})} $B$G$"$k(B.
801: @item
802: @code{sm1.deRham} $B$O7W;;5!$N;q8;$rBgNL$K;HMQ$9$k(B.
803: $B$?$H$($P(B @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
804: $B$N7W;;$9$i$9$G$KHs>o$KBgJQ$G$"$k(B.
805: @item
806: b-$B4X?t$N:,$r8zN($h$/2r@O$9$k$K$O(B, @code{ox_asir} $B$,(B @code{ox_sm1_forAsir}
807: $B$h$j;HMQ$5$l$k$Y$-$G$"$k(B. $B%3%^%s%I(B @*
808: @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
809: $B$rMQ$$$F(B, @code{ox_asir} $B$H$NDL?.%b%8%e!<%k$r$"$i$+$8$a%m!<%I$7$F$*$/$H$h$$(B.
810: $B$3$N%3%^%s%I$O(B @code{ox_asir_forAsir} $B$N%9%?!<%H;~$K<+F0E*$K<B9T$5$l$F$$$k(B.
811: @item
812: @code{sm1.deRham} $B$r(B @code{ox_reset(sm1.get_Sm1_proc());} $B$GCfCG$9$k$H(B,
813: $B0J8e(B sm1 $B%5!<%P$,HsI8=`%b!<%I$KF~$jM=4|$7$J$$F0:n$r$9$k>l9g(B
814: $B$,$"$k$N$G(B, $B%3%^%s%I(B @code{ox_shutdown(sm1.get_Sm1_proc());} $B$G(B, @code{ox_sm1_forAsir}
815: $B$r0l;~(B shutdown $B$7$F%j%9%?!<%H$7$?J}$,0BA4$G$"$k(B.
816: @end itemize
817: */
818: /*&C
819: @example
820: [332] sm1.deRham([x^3-y^2,[x,y]]);
821: [1,1,0]
822: [333] sm1.deRham([x*(x-1),[x]]);
823: [1,2]
824: @end example
825: */
826: /*&en
827: @table @t
828: @item Reference
829: @code{sm1.start}, @code{deRham} (sm1 command)
830: @item Algorithm:
831: Oaku, Takayama, An algorithm for de Rham cohomology groups of the
832: complement of an affine variety via D-module computation,
833: Journal of pure and applied algebra 139 (1999), 201--233.
834: @end table
835: */
836: /*&ja
837: @table @t
838: @item $B;2>H(B
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:
847:
848:
849:
850: /*&en
851: @c sort-sm1.hilbert
852: @node sm1.hilbert,,, SM1 Functions
853: @subsection @code{sm1.hilbert}
854: @findex sm1.hilbert
855: @findex hilbert_polynomial
856: @table @t
857: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
858: :: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
859: @item hilbert_polynomial(@var{f},@var{v})
860: :: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
861: @end table
862:
863: @table @var
864: @item return
865: Polynomial
866: @item p
867: Number
868: @item f, v
869: List
870: @end table
871:
872: @itemize @bullet
873: @item It returns the Hilbert polynomial h(k) of the set of polynomials
874: @var{f}
875: with respect to the set of variables @var{v}.
876: @item
877: h(k) = dim_Q F_k/I \cap F_k where F_k the set of polynomials of which
878: degree is less than or equal to k and I is the ideal generated by the
879: set of polynomials @var{f}.
880: @item
881: Note for sm1.hilbert:
882: For an efficient computation, it is preferable that
883: the set of polynomials @var{f} is a set of monomials.
884: In fact, this function firstly compute a Grobner basis of @var{f}, and then
885: compute the Hilbert polynomial of the initial monomials of the basis.
886: If the input @var{f} is already a Grobner
887: basis, a Grobner basis is recomputed in this function,
888: which is a waste of time and Grobner basis computation in the ring of
889: polynomials in @code{sm1} is slower than in @code{asir}.
890: @end itemize
891: */
892: /*&ja
893: @c sort-sm1.hilbert
894: @node sm1.hilbert,,, SM1 Functions
895: @subsection @code{sm1.hilbert}
896: @findex sm1.hilbert
897: @findex hilbert_polynomial
898: @table @t
899: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
900: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
901: @item hilbert_polynomial(@var{f},@var{v})
902: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
903: @end table
904:
905: @table @var
906: @item return
907: $BB?9`<0(B
908: @item p
909: $B?t(B
910: @item f, v
911: $B%j%9%H(B
912: @end table
913:
914: @itemize @bullet
915: @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)
916: $B$r7W;;$9$k(B.
917: @item
918: 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
919: $BB?9`<0$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.
920: @item
921: sm1.hilbert $B$K$+$s$9$k%N!<%H(B:
922: $B8zN($h$/7W;;$9$k$K$O(B @var{f} $B$O%b%N%_%"%k$N=89g$K$7$?J}$,$$$$(B.
923: $B<B:](B, $B$3$NH!?t$O$^$:(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$7(B, $B$=$l$+$i$=$N(B initial
924: monomial $BC#$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
925: $B$7$?$,$C$F(B, $BF~NO(B @var{f} $B$,$9$G$K%0%l%V%J4pDl$@$H$3$NH!?t$N$J$+$G$b$&0lEY(B
926: $B%0%l%V%J4pDl$N7W;;$,$*$3$J$o$l$k(B. $B$3$l$O;~4V$NL5BL$G$"$k$7(B, @code{sm1} $B$N(B
927: $BB?9`<0%0%l%V%J4pDl7W;;$O(B @code{asir} $B$h$jCY$$(B.
928: @end itemize
929: */
930:
931: /*&C
932: @example
933:
934: [346] load("katsura")$
935: [351] A=hilbert_polynomial(katsura(5),[u0,u1,u2,u3,u4,u5]);
936: 32
937:
938: @end example
939:
940: @example
941: [279] load("katsura")$
942: [280] A=gr(katsura(5),[u0,u1,u2,u3,u4,u5],0)$
943: [281] dp_ord();
944: 0
945: [282] B=map(dp_ht,map(dp_ptod,A,[u0,u1,u2,u3,u4,u5]));
946: [(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>>,
947: (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>>,
948: (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>>,
949: (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>>,
950: (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>>,
951: (1)*<<0,1,0,0,0,4>>,(1)*<<0,0,0,0,0,6>>]
952: [283] C=map(dp_dtop,B,[u0,u1,u2,u3,u4,u5]);
953: [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,
954: 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,
955: u5^4*u2,u5^4*u1,u5^6]
956: [284] sm1.hilbert([C,[u0,u1,u2,u3,u4,u5]]);
957: 32
958: @end example
959: */
960:
961: /*&en
962: @table @t
963: @item Reference
964: @code{sm1.start}, @code{sm1.gb}, @code{longname}
965: @end table
966: */
967: /*&ja
968: @table @t
969: @item $B;2>H(B
970: @code{sm1.start}, @code{sm1.gb}, @code{longname}
971: @end table
972: */
973:
974:
975: /*&en
976: @c sort-sm1.genericAnn
977: @node sm1.genericAnn,,, SM1 Functions
978: @subsection @code{sm1.genericAnn}
979: @findex sm1.genericAnn
980: @table @t
981: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
982: :: It computes the annihilating ideal for @var{f}^s.
983: @var{v} is the list of variables. Here, s is @var{v}[0] and
984: @var{f} is a polynomial in the variables @code{rest}(@var{v}).
985: @end table
986:
987: @table @var
988: @item return
989: List
990: @item p
991: Number
992: @item f
993: Polynomial
994: @item v
995: List
996: @end table
997:
998: @itemize @bullet
999: @item This function computes the annihilating ideal for @var{f}^s.
1000: @var{v} is the list of variables. Here, s is @var{v}[0] and
1001: @var{f} is a polynomial in the variables @code{rest}(@var{v}).
1002: @end itemize
1003: */
1004: /*&ja
1005: @c sort-sm1.genericAnn
1006: @node sm1.genericAnn,,, SM1 Functions
1007: @subsection @code{sm1.genericAnn}
1008: @findex sm1.genericAnn
1009: @table @t
1010: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
1011: :: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
1012: @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,
1013: @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
1014: @end table
1015:
1016: @table @var
1017: @item return
1018: $B%j%9%H(B
1019: @item p
1020: $B?t(B
1021: @item f
1022: $BB?9`<0(B
1023: @item v
1024: $B%j%9%H(B
1025: @end table
1026:
1027: @itemize @bullet
1028: @item $B$3$NH!?t$O(B,
1029: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
1030: @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,
1031: @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
1032: @end itemize
1033: */
1034: /*&C
1035: @example
1036: [595] sm1.genericAnn([x^3+y^3+z^3,[s,x,y,z]]);
1037: [-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]
1038: @end example
1039: */
1040: /*&en
1041: @table @t
1042: @item Reference
1043: @code{sm1.start}
1044: @end table
1045: */
1046: /*&ja
1047: @table @t
1048: @item $B;2>H(B
1049: @code{sm1.start}
1050: @end table
1051: */
1052:
1053:
1054:
1055: /*&en
1056: @c sort-sm1.wTensor0
1057: @node sm1.wTensor0,,, SM1 Functions
1058: @subsection @code{sm1.wTensor0}
1059: @findex sm1.wTensor0
1060: @table @t
1061: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1062: :: It computes the D-module theoretic 0-th tensor product
1063: of @var{f} and @var{g}.
1064: @end table
1065:
1066: @table @var
1067: @item return
1068: List
1069: @item p
1070: Number
1071: @item f, g, v, w
1072: List
1073: @end table
1074:
1075: @itemize @bullet
1076: @item
1077: It returns the D-module theoretic 0-th tensor product
1078: of @var{f} and @var{g}.
1079: @item
1080: @var{v} is a list of variables.
1081: @var{w} is a list of weights. The integer @var{w}[i] is
1082: the weight of the variable @var{v}[i].
1083: @item
1084: @code{sm1.wTensor0} calls @code{wRestriction0} of @code{ox_sm1},
1085: which requires a generic weight
1086: vector @var{w} to compute the restriction.
1087: If @var{w} is not generic, the computation fails.
1088: @item Let F and G be solutions of @var{f} and @var{g} respectively.
1089: Intuitively speaking, the 0-th tensor product is a system of
1090: differential equations which annihilates the function FG.
1091: @item The answer is a submodule of a free module D^r in general even if
1092: the inputs @var{f} and @var{g} are left ideals of D.
1093: @end itemize
1094: */
1095:
1096: /*&ja
1097: @c sort-sm1.wTensor0
1098: @node sm1.wTensor0,,, SM1 Functions
1099: @subsection @code{sm1.wTensor0}
1100: @findex sm1.wTensor0
1101: @table @t
1102: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1103: :: @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
1104: $B7W;;$9$k(B.
1105: @end table
1106:
1107: @table @var
1108: @item return
1109: $B%j%9%H(B
1110: @item p
1111: $B?t(B
1112: @item f, g, v, w
1113: $B%j%9%H(B
1114: @end table
1115:
1116: @itemize @bullet
1117: @item
1118: @var{f} $B$H(B @var{g} $B$N(B
1119: D-$B2C72$H$7$F$N(B 0 $B<!%F%s%=%k@Q$r7W;;$9$k(B.
1120: @item
1121: @var{v} $B$OJQ?t$N%j%9%H$G$"$k(B.
1122: @var{w} $B$O(B weight $B$N%j%9%H$G$"$k(B.
1123: $B@0?t(B @var{w}[i] $B$OJQ?t(B @var{v}[i] $B$N(B weight $B$G$"$k(B.
1124: @item
1125: @code{sm1.wTensor0} $B$O(B @code{ox_sm1} $B$N(B @code{wRestriction0}
1126: $B$r$h$s$G$$$k(B.
1127: @code{wRestriction0} $B$O(B, generic $B$J(B weight $B%Y%/%H%k(B @var{w}
1128: $B$r$b$H$K$7$F@)8B$r7W;;$7$F$$$k(B.
1129: Weight $B%Y%/%H%k(B @var{w} $B$,(B generic $B$G$J$$$H7W;;$,%(%i!<$GDd;_$9$k(B.
1130: @item F $B$*$h$S(B G $B$r(B @var{f} $B$H(B @var{g} $B$=$l$>$l$N2r$H$9$k(B.
1131: $BD>4QE*$K$$$($P(B, 0 $B<!$N%F%s%=%k@Q$O(B $B4X?t(B FG $B$N$_$?$9HyJ,J}Dx<07O$G$"$k(B.
1132: @item $BF~NO(B @var{f}, @var{g} $B$,(B D $B$N:8%$%G%"%k$G$"$C$F$b(B,
1133: $B0lHL$K(B, $B=PNO$O<+M32C72(B D^r $B$NItJ,2C72$G$"$k(B.
1134: @end itemize
1135: */
1136: /*&C
1137: @example
1138: [258] sm1.wTensor0([[x*dx -1, y*dy -4],[dx+dy,dx-dy^2],[x,y],[1,2]]);
1139: [[-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],
1140: [-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],
1141: [y^2*dy^2+(-y^2-8*y)*dy+4*y+20]]
1142: @end example
1143: */
1144:
1145:
1146:
1147: /*&en
1148: @c sort-sm1.reduction
1149: @node sm1.reduction,,, SM1 Functions
1150: @subsection @code{sm1.reduction}
1151: @findex sm1.reduction
1152: @table @t
1153: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1154: ::
1155: @end table
1156:
1157: @table @var
1158: @item return
1159: List
1160: @item f
1161: Polynomial
1162: @item g, v, w
1163: List
1164: @item p
1165: Number (the process number of ox_sm1)
1166: @end table
1167:
1168: @itemize @bullet
1169: @item It reduces @var{f} by the set of polynomial @var{g}
1170: in the homogenized Weyl algebra; it applies the
1171: division algorithm to @var{f}. The set of variables is @var{v} and
1172: @var{w} is weight vectors to determine the order, which can be ommited.
1173: @code{sm1.reduction_noH} is for the Weyl algebra.
1174: @item The return value is of the form
1175: [r,c0,[c1,...,cm],[g1,...gm]] where @var{g}=[g1, ..., gm] and
1176: c0 f + c1 g1 + ... + cm gm = r.
1177: r/c0 is the normal form.
1178: @item The function reduction reduces reducible terms that appear
1179: in lower order terms.
1180: @item The functions
1181: sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G)
1182: are for distributed polynomials.
1183: @end itemize
1184: */
1185: /*&ja
1186: @node sm1.reduction,,, SM1 Functions
1187: @subsection @code{sm1.reduction}
1188: @findex sm1.reduction
1189: @table @t
1190: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1191: ::
1192: @end table
1193:
1194: @table @var
1195: @item return
1196: $B%j%9%H(B
1197: @item f
1198: $BB?9`<0(B
1199: @item g, v, w
1200: $B%j%9%H(B
1201: @item p
1202: $B?t(B (ox_sm1 $B$N%W%m%;%9HV9f(B)
1203: @end table
1204:
1205: @itemize @bullet
1206: @item $B$3$NH!?t$O(B @var{f} $B$r(B homogenized $B%o%$%kBe?t$K$*$$$F(B,
1207: $BB?9`<0=89g(B @var{g} $B$G4JC12=(B (reduce) $B$9$k(B; $B$D$^$j(B,
1208: $B$3$NH!?t$O(B, @var{f} $B$K3d;;%"%k%4%j%:%`$rE,MQ$9$k(B.
1209: $BJQ?t=89g$O(B @var{v} $B$G;XDj$9$k(B.
1210: @var{w} $B$O=g=x$r;XDj$9$k$?$a$N(B $B%&%(%$%H%Y%/%H%k$G$"$j(B,
1211: $B>JN,$7$F$b$h$$(B.
1212: @code{sm1.reduction_noH} $B$O(B, Weyl algebra $BMQ(B.
1213: @item $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
1214: [r,c0,[c1,...,cm],g] $B$3$3$G(B @var{g}=[g1, ..., gm] $B$G$"$j(B,
1215: c0 f + c1 g1 + ... + cm gm = r
1216: $B$,$J$j$?$D(B.
1217: r/c0 $B$,(B normal form $B$G$"$k(B.
1218: @item $B$3$NH!?t$O(B, $BDc<!9`$K$"$i$o$l$k(B reducible $B$J9`$b4JC12=$9$k(B.
1219: @item $BH!?t(B
1220: sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_d(P,F,G)
1221: $B$O(B, $BJ,;6B?9`<0MQ$G$"$k(B.
1222: @end itemize
1223: */
1224: /*&C
1225: @example
1226: [259] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y]]);
1227: [x^2+y^2-4,1,[0,0],[y^4-4*y^2+1,x+y^3-4*y]]
1228: [260] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y],[[x,1]]]);
1229: [0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]]
1230: @end example
1231: */
1232: /*&en
1233: @table @t
1234: @item Reference
1235: @code{sm1.start}, @code{d_true_nf}
1236: @end table
1237: */
1238: /*&ja
1239: @table @t
1240: @item $B;2>H(B
1241: @code{sm1.start}, @code{d_true_nf}
1242: @end table
1243: */
1244:
1245:
1246: /*&en
1247: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
1248: @subsection @code{sm1.xml_tree_to_prefix_string}
1249: @findex sm1.xml_tree_to_prefix_string
1250: @table @t
1251: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1252: :: Translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
1253: @end table
1254:
1255: @table @var
1256: @item return
1257: String
1258: @item p
1259: Number
1260: @item s
1261: String
1262: @end table
1263:
1264: @itemize @bullet
1265: @item It translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
1266: @item This function should be moved to om_* in a future.
1267: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} returns CMO_TREE.
1268: asir has not yet understood this CMO.
1269: @item @code{java} execution environment is required.
1270: (For example, @code{/usr/local/jdk1.1.8/bin} should be in the
1271: command search path.)
1272: @end itemize
1273: */
1274: /*&ja
1275: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
1276: @subsection @code{sm1.xml_tree_to_prefix_string}
1277: @findex sm1.xml_tree_to_prefix_string
1278: @table @t
1279: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1280: :: XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
1281: @end table
1282:
1283: @table @var
1284: @item return
1285: String
1286: @item p
1287: Number
1288: @item s
1289: String
1290: @end table
1291:
1292: @itemize @bullet
1293: @item XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
1294: @item $B$3$NH!?t$O(B om_* $B$K>-Mh0\$9$Y$-$G$"$k(B.
1295: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} $B$O(B CMO_TREE
1296: $B$rLa$9(B. @code{asir} $B$O$3$N(B CMO $B$r$^$@%5%]!<%H$7$F$$$J$$(B.
1297: @item @code{java} $B$N<B9T4D6-$,I,MW(B.
1298: ($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.)
1299: @end itemize
1300: */
1301: /*&C
1302: @example
1303: [263] load("om");
1304: 1
1305: [270] F=om_xml(x^4-1);
1306: control: wait OX
1307: Trying to connect to the server... Done.
1308: <OMOBJ><OMA><OMS name="plus" cd="basic"/><OMA>
1309: <OMS name="times" cd="basic"/><OMA>
1310: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>4</OMI></OMA>
1311: <OMI>1</OMI></OMA><OMA><OMS name="times" cd="basic"/><OMA>
1312: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>0</OMI></OMA>
1313: <OMI>-1</OMI></OMA></OMA></OMOBJ>
1314: [271] sm1.xml_tree_to_prefix_string(F);
1315: basic_plus(basic_times(basic_power(x,4),1),basic_times(basic_power(x,0),-1))
1316: @end example
1317: */
1318: /*&en
1319: @table @t
1320: @item Reference
1321: @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
1322: @end table
1323: */
1324: /*&ja
1325: @table @t
1326: @item $B;2>H(B
1327: @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
1328: @end table
1329: */
1330:
1331:
1332:
1333:
1334: /*&en
1335: @c sort-sm1.syz
1336: @node sm1.syz,,, SM1 Functions
1337: @subsection @code{sm1.syz}
1338: @findex sm1.syz
1339: @findex sm1.syz_d
1340: @table @t
1341: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1342: :: computes the syzygy of @var{f} in the ring of differential
1343: operators with the variable @var{v}.
1344: @end table
1345:
1346: @table @var
1347: @item return
1348: List
1349: @item p
1350: Number
1351: @item f, v, w
1352: List
1353: @end table
1354:
1355: @itemize @bullet
1356: @item
1357: The return values is of the form
1358: [@var{s},[@var{g}, @var{m}, @var{t}]].
1359: Here @var{s} is the syzygy of @var{f} in the ring of differential
1360: operators with the variable @var{v}.
1361: @var{g} is a Groebner basis of @var{f} with the weight vector @var{w},
1362: and @var{m} is a matrix that translates the input matrix @var{f} to the Gr\"obner
1363: basis @var{g}.
1364: @var{t} is the syzygy of the Gr\"obner basis @var{g}.
1365: In summary, @var{g} = @var{m} @var{f} and
1366: @var{s} @var{f} = 0 hold as matrices.
1367: @item
1368: The weight vectors are given by @var{w}, which can be omitted.
1369: If @var{w} is not given,
1370: the graded reverse lexicographic order will be used to compute Grobner basis.
1371: @item
1372: When a non-term order is given, the Grobner basis is computed in
1373: the homogenized Weyl algebra (See Section 1.2 of the book of SST).
1374: The homogenization variable h is automatically added.
1375: @end itemize
1376: */
1377: /*&ja
1378: @c sort-sm1.syz
1379: @node sm1.syz,,, SM1 Functions
1380: @subsection @code{sm1.syz}
1381: @findex sm1.syz
1382: @findex sm1.syz_d
1383: @table @t
1384: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1385: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N(B syzygy $B$r7W;;$9$k(B.
1386: @end table
1387:
1388: @table @var
1389: @item return
1390: $B%j%9%H(B
1391: @item p
1392: $B?t(B
1393: @item f, v, w
1394: $B%j%9%H(B
1395: @end table
1396:
1397: @itemize @bullet
1398: @item
1399: $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
1400: [@var{s},[@var{g}, @var{m}, @var{t}]].
1401: $B$3$3$G(B @var{s} $B$O(B @var{f} $B$N(B @var{v} $B$rJQ?t$H$9$kHyJ,:nMQAG4D$K$*$1$k(B
1402: syzygy $B$G$"$k(B.
1403: @var{g} $B$O(B @var{f} $B$N(B weight vector @var{w} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
1404: @var{m} $B$OF~NO9TNs(B @var{f} $B$r%0%l%V%J4pDl(B
1405: @var{g} $B$XJQ49$9$k9TNs$G$"$k(B.
1406: @var{t} $B$O%0%l%V%J4pDl(B @var{g} $B$N(B syzygy $B$G$"$k(B.
1407: $B$^$H$a$k$H(B, $B<!$NEy<0$,$J$j$?$D(B:
1408: @var{g} = @var{m} @var{f} ,
1409: @var{s} @var{f} = 0.
1410: @item
1411: Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
1412: $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
1413: $B%V%l%V%J4pDl$r7W;;$9$k(B.
1414: @item
1415: 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).
1416: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
1417: @end itemize
1418: */
1419: /*&C
1420: @example
1421: [293] sm1.syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
1422: [[[y*x*dy*dx-2,-x*dx-y*dy+1]], generators of the syzygy
1423: [[[x*dx+y*dy-1],[y^2*dy^2+2]], grobner basis
1424: [[1,0],[y*dy,-1]], transformation matrix
1425: [[y*x*dy*dx-2,-x*dx-y*dy+1]]]]
1426: @end example
1427: */
1428: /*&C
1429: @example
1430: [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]]]);
1431: [[[y*x*dy*dx-1,-x^2*dx^2-x*dx-y^2*dy^2-y*dy+4]], generators of the syzygy
1432: [[[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
1433: [h^4*x*dx+y^3*dy^3+3*h^2*y^2*dy^2-3*h^4*y*dy]],
1434: [[1,0],[0,1],[y*dy,-x*dx]], transformation matrix
1435: [[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]]]]
1436: @end example
1437: */
1438:
1439:
1440:
1441: /*&en
1442: @node sm1.mul,,, SM1 Functions
1443: @subsection @code{sm1.mul}
1444: @findex sm1.mul
1445: @table @t
1446: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1447: :: ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
1448: @end table
1449:
1450: @table @var
1451: @item return
1452: Polynomial or List
1453: @item p
1454: Number
1455: @item f, g
1456: Polynomial or List
1457: @item v
1458: List
1459: @end table
1460:
1461: @itemize @bullet
1462: @item Ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
1463: @item @code{sm1.mul_h} is for homogenized Weyl algebra.
1464: @item BUG: @code{sm1.mul(p0*dp0,1,[p0])} returns
1465: @code{dp0*p0+1}.
1466: A variable order such that d-variables come after non-d-variables
1467: is necessary for the correct computation.
1468: @end itemize
1469: */
1470:
1471: /*&ja
1472: @node sm1.mul,,, SM1 Functions
1473: @subsection @code{sm1.mul}
1474: @findex sm1.mul
1475: @table @t
1476: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1477: :: sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
1478: $B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
1479: @end table
1480:
1481: @table @var
1482: @item return
1483: $BB?9`<0$^$?$O%j%9%H(B
1484: @item p
1485: $B?t(B
1486: @item f, g
1487: $BB?9`<0$^$?$O%j%9%H(B
1488: @item v
1489: $B%j%9%H(B
1490: @end table
1491:
1492: @itemize @bullet
1493: @item sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
1494: $B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
1495: @item @code{sm1.mul_h} $B$O(B homogenized Weyl $BBe?tMQ(B.
1496: @item BUG: @code{sm1.mul(p0*dp0,1,[p0])} $B$O(B
1497: @code{dp0*p0+1} $B$rLa$9(B.
1498: 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.
1499: @end itemize
1500: */
1501:
1502: /*&C
1503:
1504: @example
1505: [277] sm1.mul(dx,x,[x]);
1506: x*dx+1
1507: [278] sm1.mul([x,y],[1,2],[x,y]);
1508: x+2*y
1509: [279] sm1.mul([[1,2],[3,4]],[[x,y],[1,2]],[x,y]);
1510: [[x+2,y+4],[3*x+4,3*y+8]]
1511: @end example
1512:
1513: */
1514:
1515:
1516:
1517:
1518: /*&en
1519: @node sm1.distraction,,, SM1 Functions
1520: @subsection @code{sm1.distraction}
1521: @findex sm1.distraction
1522: @table @t
1523: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1524: :: ask the @code{sm1} server to compute the distraction of @var{f}.
1525: @end table
1526:
1527: @table @var
1528: @item return
1529: List
1530: @item p
1531: Number
1532: @item f
1533: Polynomial
1534: @item v,x,d,s
1535: List
1536: @end table
1537:
1538: @itemize @bullet
1539: @item It asks the @code{sm1} server of the descriptor number @var{p}
1540: to compute the distraction of @var{f} in the ring of differential
1541: operators with variables @var{v}.
1542: @item @var{x} is a list of x-variables and @var{d} is that of d-variables
1543: to be distracted. @var{s} is a list of variables to express the distracted @var{f}.
1544: @item Distraction is roughly speaking to replace x*dx by a single variable x.
1545: See Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations at page 68 for details.
1546: @end itemize
1547: */
1548:
1549: /*&ja
1550: @node sm1.distraction,,, SM1 Functions
1551:
1552: @subsection @code{sm1.distraction}
1553: @findex sm1.distraction
1554: @table @t
1555: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1556: :: @code{sm1} $B$K(B @var{f} $B$N(B distraction $B$r7W;;$7$F$b$i$&(B.
1557: @end table
1558:
1559: @table @var
1560: @item return
1561: $B%j%9%H(B
1562: @item p
1563: $B?t(B
1564: @item f
1565: $BB?9`<0(B
1566: @item v,x,d,s
1567: $B%j%9%H(B
1568: @end table
1569:
1570: @itemize @bullet
1571: @item $B<1JL;R(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B,
1572: @var{f} $B$N(B distraction $B$r(B @var{v} $B>e$NHyJ,:nMQAG4D$G7W;;$7$F$b$i$&(B.
1573: @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
1574: $B%j%9%H(B. Distraction $B$7$?$i(B, @var{s} $B$rJQ?t$H$7$F7k2L$rI=$9(B.
1575: @item Distraction $B$H$$$&$N$O(B x*dx $B$r(B x $B$GCV$-49$($k$3$H$G$"$k(B.
1576: $B>\$7$/$O(B Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations $B$N(B page 68 $B$r8+$h(B.
1577: @end itemize
1578: */
1579:
1580: /*&C
1581:
1582: @example
1583: [280] sm1.distraction([x*dx,[x],[x],[dx],[x]]);
1584: x
1585: [281] sm1.distraction([dx^2,[x],[x],[dx],[x]]);
1586: x^2-x
1587: [282] sm1.distraction([x^2,[x],[x],[dx],[x]]);
1588: x^2+3*x+2
1589: [283] fctr(@@);
1590: [[1,1],[x+1,1],[x+2,1]]
1591: [284] sm1.distraction([x*dx*y+x^2*dx^2*dy,[x,y],[x],[dx],[x]]);
1592: (x^2-x)*dy+x*y
1593: @end example
1594: */
1595:
1596: /*&en
1597: @table @t
1598: @item Reference
1599: @code{distraction2(sm1)},
1600: @end table
1601: */
1602:
1603: /*&ja
1604: @table @t
1605: @item $B;2>H(B
1606: @code{distraction2(sm1)},
1607: @end table
1608: */
1609:
1610:
1611:
1612: /*&en
1613: @node sm1.gkz,,, SM1 Functions
1614: @subsection @code{sm1.gkz}
1615: @findex sm1.gkz
1616: @table @t
1617: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1618: :: Returns the GKZ system (A-hypergeometric system) associated to the matrix
1619: @var{A} with the parameter vector @var{B}.
1620: @end table
1621:
1622: @table @var
1623: @item return
1624: List
1625: @item p
1626: Number
1627: @item A, B
1628: List
1629: @end table
1630:
1631: @itemize @bullet
1632: @item Returns the GKZ hypergeometric system
1633: (A-hypergeometric system) associated to the matrix
1634: @end itemize
1635: */
1636:
1637: /*&ja
1638: @node sm1.gkz,,, SM1 Functions
1639: @subsection @code{sm1.gkz}
1640: @findex sm1.gkz
1641: @table @t
1642: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1643: :: $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.
1644: @end table
1645:
1646: @table @var
1647: @item return
1648: $B%j%9%H(B
1649: @item p
1650: $B?t(B
1651: @item A, B
1652: $B%j%9%H(B
1653: @end table
1654:
1655: @itemize @bullet
1656: @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.
1657: @end itemize
1658: */
1659:
1660: /*&C
1661:
1662: @example
1663:
1664: [280] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
1665: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1666: -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1667: [x1,x2,x3,x4]]
1668:
1669: @end example
1670:
1671: */
1672:
1.2 takayama 1673: /*&en
1674: @node sm1.mgkz,,, SM1 Functions
1675: @subsection @code{sm1.mgkz}
1676: @findex sm1.mgkz
1677: @table @t
1678: @item sm1.mgkz([@var{A},@var{W},@var{B}]|proc=@var{p})
1679: :: Returns the modified GKZ system (A-hypergeometric system) associated to the matrix
1680: @var{A} and the weight @var{w} with the parameter vector @var{B}.
1681: @end table
1682:
1683: @table @var
1684: @item return
1685: List
1686: @item p
1687: Number
1688: @item A, W, B
1689: List
1690: @end table
1691:
1692: @itemize @bullet
1693: @item Returns the modified GKZ hypergeometric system
1694: (A-hypergeometric system) associated to the matrix
1695: @item http://arxiv.org/abs/0707.0043
1696: @end itemize
1697: */
1698:
1699: /*&ja
1700: @node sm1.mgkz,,, SM1 Functions
1701: @subsection @code{sm1.mgkz}
1702: @findex sm1.mgkz
1703: @table @t
1704: @item sm1.mgkz([@var{A},@var{W},@var{B}]|proc=@var{p})
1705: :: $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.
1706: @end table
1707:
1708: @table @var
1709: @item return
1710: $B%j%9%H(B
1711: @item p
1712: $B?t(B
1713: @item A, W, B
1714: $B%j%9%H(B
1715: @end table
1716:
1717: @itemize @bullet
1718: @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.
1719: @item http://arxiv.org/abs/0707.0043
1720: @end itemize
1721: */
1722:
1723: /*&C
1724:
1725: @example
1726:
1727: [280] sm1.mgkz([ [[1,2,3]], [1,2,1], [a/2]]);
1728: [[6*x3*dx3+4*x2*dx2+2*x1*dx1-a,-x4*dx4+x3*dx3+2*x2*dx2+x1*dx1,
1729: -dx2+dx1^2,-x4^2*dx3+dx1*dx2],[x1,x2,x3,x4]]
1730:
1731: Modified A-hypergeometric system for
1732: A=(1,2,3), w=(1,2,1), beta=(a/2).
1733: @end example
1734:
1735: */
1736:
1.1 takayama 1737:
1738:
1739:
1740: /*&en
1741: @node sm1.appell1,,, SM1 Functions
1742: @subsection @code{sm1.appell1}
1743: @findex sm1.appell1
1744: @table @t
1745: @item sm1.appell1(@var{a}|proc=@var{p})
1746: :: Returns the Appell hypergeometric system F_1 or F_D.
1747: @end table
1748:
1749: @table @var
1750: @item return
1751: List
1752: @item p
1753: Number
1754: @item a
1755: List
1756: @end table
1757:
1758: @itemize @bullet
1759: @item Returns the hypergeometric system for the Lauricella function
1760: F_D(a,b1,b2,...,bn,c;x1,...,xn)
1761: where @var{a} =(a,c,b1,...,bn).
1762: When n=2, the Lauricella function is called the Appell function F_1.
1763: The parameters a, c, b1, ..., bn may be rational numbers.
1764: @item It does not call sm1 function appell1. As a concequence,
1765: when parameters are rational or symbolic, this function also works
1766: as well as integral parameters.
1767: @end itemize
1768: */
1769:
1770: /*&ja
1771: @node sm1.appell1,,, SM1 Functions
1772: @subsection @code{sm1.appell1}
1773: @findex sm1.appell1
1774: @table @t
1775: @item sm1.appell1(@var{a}|proc=@var{p})
1776: :: F_1 $B$^$?$O(B F_D $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
1777: @end table
1778:
1779: @table @var
1780: @item return
1781: $B%j%9%H(B
1782: @item p
1783: $B?t(B
1784: @item a
1785: $B%j%9%H(B
1786: @end table
1787:
1788: @itemize @bullet
1789: @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
1790: F_D(a,b1,b2,...,bn,c;x1,...,xn)
1791: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
1792: @var{a} =(a,c,b1,...,bn).
1793: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
1794: @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
1795: $B@5$7$/F0$/(B.
1796: @end itemize
1797: */
1798:
1799: /*&C
1800:
1801: @example
1802:
1803: [281] sm1.appell1([1,2,3,4]);
1804: [[((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2+(-5*x1+2)*dx1-3,
1805: (-x2^2+x2)*dx2^2+((-x1*x2+x1)*dx1-6*x2+2)*dx2-4*x1*dx1-4,
1806: ((-x2+x1)*dx1+3)*dx2-4*dx1], equations
1807: [x1,x2]] the list of variables
1808:
1809: [282] sm1.gb(@@);
1810: [[((-x2+x1)*dx1+3)*dx2-4*dx1,((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2
1811: +(-5*x1+2)*dx1-3,(-x2^2+x2)*dx2^2+((-x2^2+x1)*dx1-3*x2+2)*dx2
1812: +(-4*x2-4*x1)*dx1-4,
1813: (x2^3+(-x1-1)*x2^2+x1*x2)*dx2^2+((-x1*x2+x1^2)*dx1+6*x2^2
1814: +(-3*x1-2)*x2+2*x1)*dx2-4*x1^2*dx1+4*x2-4*x1],
1815: [x1*dx1*dx2,-x1^2*dx1^2,-x2^2*dx1*dx2,-x1*x2^2*dx2^2]]
1816:
1817: [283] sm1.rank(sm1.appell1([1/2,3,5,-1/3]));
1818: 3
1819:
1820: [285] Mu=2$ Beta = 1/3$
1821: [287] sm1.rank(sm1.appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));
1822: 4
1823:
1824:
1825: @end example
1826:
1827: */
1828:
1829: /*&en
1830: @node sm1.appell4,,, SM1 Functions
1831: @subsection @code{sm1.appell4}
1832: @findex sm1.appell4
1833: @table @t
1834: @item sm1.appell4(@var{a}|proc=@var{p})
1835: :: Returns the Appell hypergeometric system F_4 or F_C.
1836: @end table
1837:
1838: @table @var
1839: @item return
1840: List
1841: @item p
1842: Number
1843: @item a
1844: List
1845: @end table
1846:
1847: @itemize @bullet
1848: @item Returns the hypergeometric system for the Lauricella function
1849: F_4(a,b,c1,c2,...,cn;x1,...,xn)
1850: where @var{a} =(a,b,c1,...,cn).
1851: When n=2, the Lauricella function is called the Appell function F_4.
1852: The parameters a, b, c1, ..., cn may be rational numbers.
1853: @item @item It does not call sm1 function appell4. As a concequence,
1854: when parameters are rational or symbolic, this function also works
1855: as well as integral parameters.
1856: @end itemize
1857: */
1858:
1859: /*&ja
1860: @node sm1.appell4,,, SM1 Functions
1861: @subsection @code{sm1.appell4}
1862: @findex sm1.appell4
1863: @table @t
1864: @item sm1.appell4(@var{a}|proc=@var{p})
1865: :: F_4 $B$^$?$O(B F_C $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
1866: @end table
1867:
1868: @table @var
1869: @item return
1870: $B%j%9%H(B
1871: @item p
1872: $B?t(B
1873: @item a
1874: $B%j%9%H(B
1875: @end table
1876:
1877: @itemize @bullet
1878: @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
1879: F_C(a,b,c1,c2,...,cn;x1,...,xn)
1880: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
1881: @var{a} =(a,b,c1,...,cn).
1882: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
1883: @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
1884: $B@5$7$/F0$/(B.
1885: @end itemize
1886: */
1887:
1888: /*&C
1889:
1890: @example
1891:
1892: [281] sm1.appell4([1,2,3,4]);
1893: [[-x2^2*dx2^2+(-2*x1*x2*dx1-4*x2)*dx2+(-x1^2+x1)*dx1^2+(-4*x1+3)*dx1-2,
1894: (-x2^2+x2)*dx2^2+(-2*x1*x2*dx1-4*x2+4)*dx2-x1^2*dx1^2-4*x1*dx1-2],
1895: equations
1896: [x1,x2]] the list of variables
1897:
1898: [282] sm1.rank(@@);
1899: 4
1900:
1901: @end example
1902:
1903: */
1904:
1905:
1906:
1907:
1908: /*&en
1909: @node sm1.rank,,, SM1 Functions
1910: @subsection @code{sm1.rank}
1911: @findex sm1.rank
1912: @table @t
1913: @item sm1.rank(@var{a}|proc=@var{p})
1914: :: Returns the holonomic rank of the system of differential equations @var{a}.
1915: @end table
1916:
1917: @table @var
1918: @item return
1919: Number
1920: @item p
1921: Number
1922: @item a
1923: List
1924: @end table
1925:
1926: @itemize @bullet
1927: @item It evaluates the dimension of the space of holomorphic solutions
1928: at a generic point of the system of differential equations @var{a}.
1929: The dimension is called the holonomic rank.
1930: @item @var{a} is a list consisting of a list of differential equations and
1931: a list of variables.
1932: @item @code{sm1.rrank} returns the holonomic rank when @var{a} is regular
1933: holonomic. It is generally faster than @code{sm1.rank}.
1934: @end itemize
1935: */
1936:
1937: /*&ja
1938: @node sm1.rank,,, SM1 Functions
1939: @subsection @code{sm1.rank}
1940: @findex sm1.rank
1941: @table @t
1942: @item sm1.rank(@var{a}|proc=@var{p})
1943: :: $BHyJ,J}Dx<07O(B @var{a} $B$N(B holonomic rank $B$rLa$9(B.
1944: @end table
1945:
1946: @table @var
1947: @item return
1948: $B?t(B
1949: @item p
1950: $B?t(B
1951: @item a
1952: $B%j%9%H(B
1953: @end table
1954:
1955: @itemize @bullet
1956: @item $BHyJ,J}Dx<07O(B @var{a} $B$N(B, generic point $B$G$N@5B'2r$N<!85$r(B
1957: $BLa$9(B. $B$3$N<!85$r(B, holonomic rank $B$H8F$V(B.
1958: @item @var{a} $B$OHyJ,:nMQAG$N%j%9%H$HJQ?t$N%j%9%H$h$j$J$k(B.
1959: @item @var{a} $B$,(B regular holonomic $B$N$H$-$O(B @code{sm1.rrank}
1960: $B$b(B holonomic rank $B$rLa$9(B.
1961: $B$$$C$Q$s$K$3$N4X?t$NJ}$,(B @code{sm1.rank} $B$h$jAa$$(B.
1962: @end itemize
1963: */
1964:
1965: /*&C
1966:
1967: @example
1968:
1969: [284] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
1970: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1971: -dx1*dx4+dx2*dx3, -dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1972: [x1,x2,x3,x4]]
1973: [285] sm1.rrank(@@);
1974: 4
1975:
1976: [286] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [1,2]]);
1977: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1978: -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1979: [x1,x2,x3,x4]]
1980: [287] sm1.rrank(@@);
1981: 5
1982:
1983: @end example
1984:
1985: */
1986:
1987:
1988: /*&en
1989: @node sm1.auto_reduce,,, SM1 Functions
1990: @subsection @code{sm1.auto_reduce}
1991: @findex sm1.auto_reduce
1992: @table @t
1993: @item sm1.auto_reduce(@var{s}|proc=@var{p})
1994: :: Set the flag "AutoReduce" to @var{s}.
1995: @end table
1996:
1997: @table @var
1998: @item return
1999: Number
2000: @item p
2001: Number
2002: @item s
2003: Number
2004: @end table
2005:
2006: @itemize @bullet
2007: @item If @var{s} is 1, then all Grobner bases to be computed
2008: will be the reduced Grobner bases.
2009: @item If @var{s} is 0, then Grobner bases are not necessarily the reduced
2010: Grobner bases. This is the default.
2011: @end itemize
2012: */
2013:
2014: /*&ja
2015: @node sm1.auto_reduce,,, SM1 Functions
2016: @subsection @code{sm1.auto_reduce}
2017: @findex sm1.auto_reduce
2018: @table @t
2019: @item sm1.auto_reduce(@var{s}|proc=@var{p})
2020: :: $B%U%i%0(B "AutoReduce" $B$r(B @var{s} $B$K@_Dj(B.
2021: @end table
2022:
2023: @table @var
2024: @item $BLa$jCM(B
2025: $B?t(B
2026: @item p
2027: $B?t(B
2028: @item s
2029: $B?t(B
2030: @end table
2031:
2032: @itemize @bullet
2033: @item @var{s} $B$,(B 1 $B$N$H$-(B, $B0J8e7W;;$5$l$k%0%l%V%J4pDl$O$9$Y$F(B,
2034: reduced $B%0%l%V%J4pDl$H$J$k(B.
2035: @item @var{s} $B$,(B 0 $B$N$H$-(B, $B7W;;$5$l$k%0%l%V%J4pDl$O(B
2036: reduced $B%0%l%V%J4pDl$H$O$+$.$i$J$$(B. $B$3$A$i$,%G%U%)!<%k%H(B.
2037: @end itemize
2038: */
2039:
2040:
2041:
2042: /*&en
2043: @node sm1.slope,,, SM1 Functions
2044: @subsection @code{sm1.slope}
2045: @findex sm1.slope
2046: @table @t
2047: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
2048: :: Returns the slopes of differential equations @var{ii}.
2049: @end table
2050:
2051: @table @var
2052: @item return
2053: List
2054: @item p
2055: Number
2056: @item ii
2057: List (equations)
2058: @item v
2059: List (variables)
2060: @item f_filtration
2061: List (weight vector)
2062: @item v_filtration
2063: List (weight vector)
2064: @end table
2065:
2066: @itemize @bullet
2067: @item @code{sm1.slope} returns the (geometric) slopes
2068: of the system of differential equations @var{ii}
2069: along the hyperplane specified by
2070: the V filtration @var{v_filtration}.
2071: @item @var{v} is a list of variables.
2072: @item The return value is a list of lists.
2073: The first entry of each list is the slope and the second entry
2074: is the weight vector for which the microcharacteristic variety is
2075: not bihomogeneous.
2076: @end itemize
2077:
2078: @noindent
2079: Algorithm:
2080: see "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
2081: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
1.4 takayama 2082: Note that the signs of the slopes s' are negative, but the absolute values -s'
1.1 takayama 2083: of the slopes are returned.
1.4 takayama 2084: In other words, when pF+qV is the gap, -s'=q/p is returned.
2085: Note that s=1-1/s' is called the slope in recent literatures. Solutions belongs to O(s).
2086: The number s satisfies 1<= s.
2087: 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)
2088: in the Borel and Laplace transformations respectively.
1.1 takayama 2089:
2090: */
2091:
2092: /*&ja
2093: @node sm1.slope,,, SM1 Functions
2094: @subsection @code{sm1.slope}
2095: @findex sm1.slope
2096: @table @t
2097: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
2098: :: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B slope $B$rLa$9(B.
2099: @end table
2100:
2101: @table @var
2102: @item return
2103: $B?t(B
2104: @item p
2105: $B?t(B
2106: @item ii
2107: $B%j%9%H(B ($BJ}Dx<0(B)
2108: @item v
2109: $B%j%9%H(B ($BJQ?t(B)
2110: @item f_filtration
2111: $B%j%9%H(B (weight vector)
2112: @item v_filtration
2113: $B%j%9%H(B (weight vector)
2114: @end table
2115:
2116: @itemize @bullet
2117: @item @code{sm1.slope} $B$O(B
2118: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B V filtration @var{v_filtration}
2119: $B$G;XDj$9$kD6J?LL$K1h$C$F$N(B (geomeric) slope $B$r7W;;$9$k(B.
2120: @item @var{v} $B$OJQ?t$N%j%9%H(B.
2121: @item $BLa$jCM$O(B, $B%j%9%H$r@.J,$H$9$k%j%9%H$G$"$k(B.
2122: $B@.J,%j%9%H$NBh(B 1 $BMWAG$,(B slope, $BBh(B 2 $BMWAG$O(B, $B$=$N(B weight vector $B$KBP1~$9$k(B
2123: microcharacteristic variety $B$,(B bihomogeneous $B$G$J$$(B.
2124: @end itemize
2125:
2126: @noindent
2127: Algorithm:
2128: "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
2129: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
2130: $B$r$_$h(B.
1.4 takayama 2131: Slope s' $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B, $B$3$N%W%m%0%i%`$O(B,
2132: Slope $B$N@dBPCM(B -s' $B$rLa$9(B.
2133: $B$D$^$j(B pF+qV $B$,(Bmicro$BFC@-B?MMBN$N(Bgap$B$G$"$k$H$-(B, -s'=q/p $B$rLa$9(B.
2134: $B:G6a$NJ88%$G$O(B s=1-1/s' $B$r(B slope $B$H8F$s$G$$$k(B. $B2r$O(B O(s) $B$KB0$9$k(B.
2135: $B?t(B s $B$O(B 1<= s $B$rK~$9(B.
2136: r=s-1=-1/s' $B$*$h$S(B kappa=1/r=-s' $B$G$"$k(B.
2137: $B$3$l$i$N?t$O(BBorel and Laplace $BJQ49$K$*$$$F$=$l$>$l(B 1/Gamma(1+m*r) factor,
2138: exp(-tau^kappa) $B9`$H$7$F;H$o$l$k(B.
1.1 takayama 2139: */
2140:
2141: /*&C
2142:
2143: @example
2144:
2145: [284] A= sm1.gkz([ [[1,2,3]], [-3] ]);
2146:
2147:
2148: [285] sm1.slope(A[0],A[1],[0,0,0,1,1,1],[0,0,-1,0,0,1]);
2149:
2150: [286] A2 = sm1.gkz([ [[1,1,1,0],[2,-3,1,-3]], [1,0]]);
2151: (* This is an interesting example given by Laura Matusevich,
2152: June 9, 2001 *)
2153:
2154: [287] sm1.slope(A2[0],A2[1],[0,0,0,0,1,1,1,1],[0,0,0,-1,0,0,0,1]);
2155:
2156:
2157: @end example
2158:
2159: */
2160: /*&en
2161: @table @t
2162: @item Reference
2163: @code{sm.gb}
2164: @end table
2165: */
2166: /*&ja
2167: @table @t
2168: @item $B;2>H(B
2169: @code{sm.gb}
2170: @end table
2171: */
2172:
2173:
2174: /*&en
2175: @include sm1-auto.en
2176: */
2177:
2178: /*&ja
2179: @include sm1-auto.ja
2180: */
2181:
2182:
2183: end$
2184:
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