Annotation of OpenXM/src/asir-contrib/packages/doc/sm1.oxweave, Revision 1.13
1.13 ! takayama 1: /*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1.oxweave,v 1.12 2003/07/28 01:17:39 takayama Exp $ */
1.1 takayama 2:
3: /*&C-texi
4: @c DO NOT EDIT THIS FILE oxphc.texi
5: */
1.6 takayama 6: /*&C-texi
7: @node SM1 Functions,,, Top
1.11 takayama 8:
1.6 takayama 9: */
1.1 takayama 10: /*&jp-texi
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} $B$GDj5A$5$l$F$$$k(B.
16: @file{sm1} $B$O(B @file{$(OpenXM_HOME)/lib/asir-contrib} $B$K$"$k(B.
17: $B%7%9%F%`(B @code{sm1} $B$OHyJ,:nMQAG4D$G7W;;$9$k$?$a$N%7%9%F%`$G$"$k(B.
18: $B7W;;Be?t4v2?$N$$$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:
21: $B$H$3$KCG$j$,$J$$$+$.$j$3$N@a$N$9$Y$F$N4X?t$O(B,
22: $BM-M}?t78?t$N<0$rF~NO$H$7$F$&$1$D$1$J$$(B.
23: $B$9$Y$F$NB?9`<0$N78?t$O@0?t$G$J$$$H$$$1$J$$(B.
24:
25: @tex
26: $B6u4V(B
27: $X:={\bf C} \setminus \{ 0, 1 \} = {\bf C} \setminus V(x(x-1))$
28: $B$N%I%i!<%`%3%[%b%m%872C#$N<!85$r7W;;$7$F$_$h$&(B.
29: $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
30: $B$^$o$kFs$D$N%k!<%W$,(B1$B<!85$N%[%b%m%8!<72$N6u4V$r$O$k(B.
31: $B$7$?$,$C$F(B, 1$B<!85%I%i!<%`%3%[%b%m%872$N<!85$O(B $2$ $B$G$"$k(B.
32: @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
33: $B<!85$rEz$($k(B.
34: @end tex
35: */
36: /*&eg-texi
37: @chapter SM1 Functions
38:
39: This chapter describes interface functions for
40: sm1 ox server @code{ox_sm1_forAsir}.
41: These interface functions are defined in the file @file{sm1}.
42: The file @file{sm1} is @*
43: at @file{$(OpenXM_HOME)/lib/asir/contrib-packages}.
44: The system @code{sm1} is a system to compute in the ring of differential
45: operators.
46: Many constructions of invariants
47: in the computational algebraic geometry reduce
48: to constructions in the ring of differential operators.
49: Documents on @code{sm1} are in
50: the directory @code{OpenXM/doc/kan96xx}.
51:
52: All the coefficients of input polynomials should be
53: integers for most functions in this section.
54: Other functions accept rational numbers as inputs
55: and it will be explicitely noted in each explanation
56: of these functions.
57:
58:
59:
60: @tex
61: Let us evaluate the dimensions of the de Rham cohomology groups
62: of
63: $X:={\bf C} \setminus \{ 0, 1 \} = {\bf C} \setminus V(x(x-1))$.
64: The space $X$ is a two punctured plane, so two loops that encircles the
65: points $x=0$ and $x=1$ respectively spans the first homology group.
66: Hence, the dimension of the first de Rham cohomology group is $2$.
67: @code{sm1} answers the dimensions of the 0th and the first
68: cohomology groups.
69: @end tex
70: */
71: /*&C-texi
72: @example
73:
1.5 takayama 74: @include opening.texi
1.1 takayama 75:
1.8 takayama 76: [283] sm1.deRham([x*(x-1),[x]]);
1.1 takayama 77: [1,2]
78: @end example
79: */
80: /*&C-texi
81: @noindent
82: The author of @code{sm1} : Nobuki Takayama, @code{takayama@@math.sci.kobe-u.ac.jp} @*
83: The author of sm1 packages : Toshinori Oaku, @code{oaku@@twcu.ac.jp} @*
84: Reference: [SST] Saito, M., Sturmfels, B., Takayama, N.,
85: Grobner Deformations of Hypergeometric Differential Equations,
86: 1999, Springer.
87: See the appendix.
88: */
1.6 takayama 89:
1.11 takayama 90: /*&C-texi
1.6 takayama 91: @menu
92: * ox_sm1_forAsir::
1.8 takayama 93: * sm1.start::
1.10 takayama 94: * sm1.sm1::
1.8 takayama 95: * sm1.push_int0::
96: * sm1.gb::
97: * sm1.deRham::
98: * sm1.hilbert::
99: * sm1.genericAnn::
100: * sm1.wTensor0::
101: * sm1.reduction::
102: * sm1.xml_tree_to_prefix_string::
103: * sm1.syz::
104: * sm1.mul::
105: * sm1.distraction::
106: * sm1.gkz::
107: * sm1.appell1::
108: * sm1.appell4::
109: * sm1.rank::
110: * sm1.auto_reduce::
111: * sm1.slope::
1.11 takayama 112: * sm1.ahg::
113: * sm1.bfunction::
114: * sm1.generalized_bfunction::
115: * sm1.restriction::
116: * sm1.saturation::
1.6 takayama 117: @end menu
118: */
119:
1.1 takayama 120: /*&jp-texi
121: @section @code{ox_sm1_forAsir} $B%5!<%P(B
122: */
123: /*&eg-texi
124: @section @code{ox_sm1_forAsir} Server
125: */
126:
127: /*&eg-texi
1.11 takayama 128: @node ox_sm1_forAsir,,, SM1 Functions
1.1 takayama 129: @subsection @code{ox_sm1_forAsir}
130: @findex ox_sm1_forAsir
131: @table @t
132: @item ox_sm1_forAsir
133: :: @code{sm1} server for @code{asir}.
134: @end table
135: @itemize @bullet
136: @item
137: @code{ox_sm1_forAsir} is the @code{sm1} server started from asir
1.8 takayama 138: by the command @code{sm1.start}.
1.1 takayama 139: In the standard setting, @*
140: @code{ox_sm1_forAsir} =
141: @file{$(OpenXM_HOME)/lib/sm1/bin/ox_sm1}
142: +
143: @file{$(OpenXM_HOME)/lib/sm1/callsm1.sm1} (macro file) @*
144: +
145: @file{$(OpenXM_HOME)/lib/sm1/callsm1b.sm1} (macro file) @*
146: The macro files @file{callsm1.sm1} and @file{callsm1b.sm1}
147: are searched from
148: current directory, @code{$(LOAD_SM1_PATH)},
149: @code{$(OpenXM_HOME)/lib/sm1},
150: @code{/usr/local/lib/sm1}
151: in this order.
152: @item Note for programmers: See the files
153: @file{$(OpenXM_HOME)/src/kxx/oxserver00.c},
154: @file{$(OpenXM_HOME)/src/kxx/sm1stackmachine.c}
155: to build your own server by reading @code{sm1} macros.
156: @end itemize
157: */
158: /*&jp-texi
1.11 takayama 159: @node ox_sm1_forAsir,,, SM1 Functions
1.1 takayama 160: @subsection @code{ox_sm1_forAsir}
161: @findex ox_sm1_forAsir
162: @table @t
163: @item ox_sm1_forAsir
164: :: @code{asir} $B$N$?$a$N(B @code{sm1} $B%5!<%P(B.
165: @end table
166: @itemize @bullet
167: @item
168: $B%5!<%P(B @code{ox_sm1_forAsir} $B$O(B @code{asir} $B$h$j%3%^%s%I(B
1.8 takayama 169: @code{sm1.start} $B$G5/F0$5$l$k(B @code{sm1} $B%5!<%P$G$"$k(B.
1.1 takayama 170:
171: $BI8=`E*@_Dj$G$O(B, @*
172: @code{ox_sm1_forAsir} =
173: @file{$(OpenXM_HOME)/lib/sm1/bin/ox_sm1}
174: +
175: @file{$(OpenXM_HOME)/lib/sm1/callsm1.sm1} (macro file) @*
176: +
177: @file{$(OpenXM_HOME)/lib/sm1/callsm1b.sm1} (macro file) @*
178: $B$G$"$j(B, $B$3$l$i$N%^%/%m%U%!%$%k$O(B, $B0lHL$K$O(B
179: current directory, @code{$(LOAD_SM1_PATH)},
180: @code{$(OpenXM_HOME)/lib/sm1},
181: @code{/usr/local/lib/sm1}
182: $B$N=gHV$G$5$,$5$l$k(B.
183: @item $B%W%m%0%i%^!<$N$?$a$N%N!<%H(B:
184: @code{sm1} $B%^%/%m$rFI$_9~$s$G<+J,FH<+$N%5!<%P$r:n$k$K$O(B
185: $B<!$N%U%!%$%k$b8+$h(B
186: @file{$(OpenXM_HOME)/src/kxx/oxserver00.c},
187: @file{$(OpenXM_HOME)/src/kxx/sm1stackmachine.c}
188: @end itemize
189: */
190:
191:
192: /*&jp-texi
193: @section $BH!?t0lMw(B
194: */
195: /*&eg-texi
196: @section Functions
197: */
198:
199: /*&eg-texi
1.8 takayama 200: @c sort-sm1.start
201: @node sm1.start,,, SM1 Functions
202: @subsection @code{sm1.start}
203: @findex sm1.start
1.1 takayama 204: @table @t
1.8 takayama 205: @item sm1.start()
1.1 takayama 206: :: Start @code{ox_sm1_forAsir} on the localhost.
207: @end table
208:
209: @table @var
210: @item return
211: Integer
212: @end table
213:
214: @itemize @bullet
215: @item Start @code{ox_sm1_forAsir} on the localhost.
216: It returns the descriptor of @code{ox_sm1_forAsir}.
217: @item Set @code{Xm_noX = 1} to start @code{ox_sm1_forAsir}
218: without a debug window.
219: @item You might have to set suitable orders of variable by the command
220: @code{ord}. For example,
221: when you are working in the
222: ring of differential operators on the variable @code{x} and @code{dx}
223: (@code{dx} stands for
224: @tex $\partial/\partial x$
225: @end tex
226: ),
227: @code{sm1} server assumes that
228: the variable @code{dx} is collected to the right and the variable
229: @code{x} is collected to the left in the printed expression.
230: In the example below, you must not use the variable @code{cc}
231: for computation in @code{sm1}.
232: @item The variables from @code{a} to @code{z} except @code{d} and @code{o}
233: and @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},
234: @code{z0}, ..., @code{z20} can be used as variables for ring of
235: differential operators in default. (cf. @code{Sm1_ord_list} in @code{sm1}).
1.8 takayama 236: @item The descriptor is stored in @code{static Sm1_proc}.
237: The descriptor can be obtained by the function
238: @code{sm1.get_Sm1_proc()}.
1.1 takayama 239: @end itemize
240: */
241: /*&jp-texi
1.8 takayama 242: @c sort-sm1.start
243: @node sm1.start,,, SM1 Functions
244: @subsection @code{sm1.start}
245: @findex sm1.start
1.1 takayama 246: @table @t
1.8 takayama 247: @item sm1.start()
1.1 takayama 248: :: localhost $B$G(B @code{ox_sm1_forAsir} $B$r%9%?!<%H$9$k(B.
249: @end table
250:
251: @table @var
252: @item return
253: $B@0?t(B
254: @end table
255:
256: @itemize @bullet
257: @item localhost $B$G(B @code{ox_sm1_forAsir} $B$r%9%?!<%H$9$k(B.
258: $B%5!<%P(B @code{ox_sm1_forAsir} $B$N<1JLHV9f$rLa$9(B.
259: @item @code{Xm_noX = 1} $B$H$*$/$H%5!<%P(B @code{ox_sm1_forAsir} $B$r%G%P%C%0MQ$N(B
260: $B%&%#%s%I%&$J$7$K5/F0$G$-$k(B.
261: @item $B%3%^%s%I(B @code{ord} $B$rMQ$$$FJQ?t=g=x$r@5$7$/@_Dj$7$F$*$/I,MW$,(B
262: $B$"$k(B.
263: $B$?$H$($P(B,
264: $BJQ?t(B @code{x} $B$H(B @code{dx} $B>e$NHyJ,:nMQAG4D(B
265: (@code{dx} $B$O(B
266: @tex $\partial/\partial x$
267: @end tex
268: $B$KBP1~(B)
269: $B$G7W;;$7$F$$$k$H$-(B,
270: @code{sm1} $B%5!<%P$O<0$r0u:~$7$?$H$-(B,
271: $BJQ?t(B @code{dx} $B$O1&B&$K=8$a$lJQ?t(B
272: @code{x} $B$O:8B&$K$"$D$a$i$l$F$$$k$H2>Dj$7$F$$$k(B.
273: $B<!$NNc$G$O(B, $BJQ?t(B @code{cc} $B$r(B @code{sm1} $B$G$N7W;;$N$?$a$KMQ$$$F$O$$$1$J$$(B.
274: @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,
275: $B$=$l$+$i(B, @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},
276: @code{z0}, ..., @code{z20} $B$O(B, $B%G%U%)!<%k%H$GHyJ,:nMQAG4D$NJQ?t$H$7$F(B
277: $B;H$($k(B (cf. @code{Sm1_ord_list} in @code{sm1}).
1.8 takayama 278: @item $B<1JLHV9f$O(B @code{static Sm1_proc} $B$K3JG<$5$l$k(B.
279: $B$3$N<1JLHV9f$O4X?t(B @code{sm1.get_Sm1_proc()} $B$G$H$j$@$9$3$H$,$G$-$k(B.
1.1 takayama 280: @end itemize
281: */
282: /*&C-texi
283: @example
284: [260] ord([da,a,db,b]);
285: [da,a,db,b,dx,dy,dz,x,y,z,dt,ds,t,s,u,v,w,
286: ......... omit ..................
287: ]
288: [261] a*da;
289: a*da
290: [262] cc*dcc;
291: dcc*cc
1.8 takayama 292: [263] sm1.mul(da,a,[a]);
1.1 takayama 293: a*da+1
1.8 takayama 294: [264] sm1.mul(a,da,[a]);
1.1 takayama 295: a*da
296: @end example
297: */
298: /*&eg-texi
299: @table @t
300: @item Reference
1.8 takayama 301: @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
1.1 takayama 302: @code{ord}
303: @end table
304: */
305: /*&jp-texi
306: @table @t
307: @item $B;2>H(B
1.8 takayama 308: @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
1.1 takayama 309: @code{ord}
310: @end table
311: */
312:
313:
314:
315: /*&eg-texi
316: @c sort-sm1
1.10 takayama 317: @node sm1.sm1,,, SM1 Functions
318: @subsection @code{sm1.sm1}
319: @findex sm1.sm1
1.1 takayama 320: @table @t
1.10 takayama 321: @item sm1.sm1(@var{p},@var{s})
1.1 takayama 322: :: ask the @code{sm1} server to execute the command string @var{s}.
323: @end table
324:
325: @table @var
326: @item return
327: Void
328: @item p
329: Number
330: @item s
331: String
332: @end table
333:
334: @itemize @bullet
335: @item It asks the @code{sm1} server of the descriptor number @var{p}
336: to execute the command string @var{s}.
1.9 takayama 337: (In the next example, the descriptor number is 0.)
1.1 takayama 338: @end itemize
339: */
340: /*&jp-texi
1.10 takayama 341: @node sm1.sm1,,, SM1 Functions
342: @subsection @code{sm1.sm1}
343: @findex sm1.sm1
1.1 takayama 344: @table @t
1.10 takayama 345: @item sm1.sm1(@var{p},@var{s})
1.1 takayama 346: :: $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.
347: @end table
348:
349: @table @var
350: @item return
351: $B$J$7(B
352: @item p
353: $B?t(B
354: @item s
355: $BJ8;zNs(B
356: @end table
357:
358: @itemize @bullet
359: @item $B<1JLHV9f(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B
360: $B%3%^%s%INs(B @var{s} $B$r<B9T$7$F$/$l$k$h$&$KMj$`(B.
1.9 takayama 361: ($B<!$NNc$G$O(B, $B<1JLHV9f(B 0)
1.1 takayama 362: @end itemize
363: */
364: /*&C-texi
365: @example
1.10 takayama 366: [261] sm1.sm1(0," ( (x-1)^2 ) . ");
1.1 takayama 367: 0
368: [262] ox_pop_string(0);
369: x^2-2*x+1
1.10 takayama 370: [263] sm1.sm1(0," [(x*(x-1)) [(x)]] deRham ");
1.1 takayama 371: 0
372: [264] ox_pop_string(0);
373: [1 , 2]
374: @end example
375: */
1.10 takayama 376:
1.1 takayama 377: /*&jp-texi
378: @table @t
379: @item $B;2>H(B
1.8 takayama 380: @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
1.1 takayama 381: @end table
382: */
383: /*&eg-texi
384: @table @t
385: @item Reference
1.8 takayama 386: @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
1.1 takayama 387: @end table
388: */
389:
390:
391: /*&eg-texi
1.8 takayama 392: @c sort-sm1.push_int0
393: @node sm1.push_int0,,, SM1 Functions
394: @subsection @code{sm1.push_int0}
395: @findex sm1.push_int0
1.1 takayama 396: @table @t
1.8 takayama 397: @item sm1.push_int0(@var{p},@var{f})
1.1 takayama 398: :: push the object @var{f} to the server with the descriptor number @var{p}.
399: @end table
400:
401: @table @var
402: @item return
403: Void
404: @item p
405: Number
406: @item f
407: Object
408: @end table
409:
410: @itemize @bullet
411: @item When @code{type(@var{f})} is 2 (recursive polynomial),
412: @var{f} is converted to a string (type == 7)
413: and is sent to the server by @code{ox_push_cmo}.
414: @item When @code{type(@var{f})} is 0 (zero),
415: it is translated to the 32 bit integer zero
416: on the server.
417: Note that @code{ox_push_cmo(@var{p},0)} sends @code{CMO_NULL} to the server.
418: In other words, the server does not get the 32 bit integer 0 nor
419: the bignum 0.
420: @item @code{sm1} integers are classfied into the 32 bit integer and
421: the bignum.
422: When @code{type(@var{f})} is 1 (number), it is translated to the
423: 32 bit integer on the server.
424: Note that @code{ox_push_cmo(@var{p},1234)} send the bignum 1234 to the
425: @code{sm1} server.
426: @item In other cases, @code{ox_push_cmo} is called without data conversion.
427: @end itemize
428: */
429: /*&jp-texi
1.8 takayama 430: @c sort-sm1.push_int0
431: @node sm1.push_int0,,, SM1 Functions
432: @subsection @code{sm1.push_int0}
433: @findex sm1.push_int0
1.1 takayama 434: @table @t
1.8 takayama 435: @item sm1.push_int0(@var{p},@var{f})
1.1 takayama 436: :: $B%*%V%8%'%/%H(B @var{f} $B$r<1JL;R(B @var{p} $B$N%5!<%P$XAw$k(B.
437: @end table
438:
439: @table @var
440: @item return
441: $B$J$7(B
442: @item p
443: $B?t(B
444: @item f
445: $B%*%V%8%'%/%H(B
446: @end table
447:
448: @itemize @bullet
449: @item @code{type(@var{f})} $B$,(B 2 ($B:F5"B?9`<0(B) $B$N$H$-(B,
450: @var{f} $B$OJ8;zNs(B (type == 7) $B$KJQ49$5$l$F(B,
451: @code{ox_push_cmo} $B$rMQ$$$F%5!<%P$XAw$i$l$k(B.
452: @item @code{type(@var{f})} $B$,(B 0 (zero) $B$N$H$-$O(B,
453: $B%5!<%P>e$G$O(B, 32 bit $B@0?t$H2r<a$5$l$k(B.
454: $B$J$*(B @code{ox_push_cmo(P,0)} $B$O%5!<%P$KBP$7$F(B @code{CMO_NULL}
455: $B$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.
456: @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.
457: @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
458: $B$*$/$k(B.
459: @code{ox_push_cmo(@var{p},1234)} $B$O(B bignum $B$N(B 1234 $B$r(B
460: @code{sm1} $B%5!<%P$K$*$/$k$3$H$KCm0U$7$h$&(B.
461: @item $B$=$NB>$N>l9g$K$O(B @code{ox_push_cmo} $B$r%G!<%?7?$NJQ49$J$7$K8F$S=P$9(B.
462: @end itemize
463: */
464: /*&C
465: @example
1.8 takayama 466: [219] P=sm1.start();
1.1 takayama 467: 0
1.8 takayama 468: [220] sm1.push_int0(P,x*dx+1);
1.1 takayama 469: 0
470: [221] A=ox_pop_cmo(P);
471: x*dx+1
472: [223] type(A);
473: 7 (string)
474: @end example
475:
476: @example
1.8 takayama 477: [271] sm1.push_int0(0,[x*(x-1),[x]]);
1.1 takayama 478: 0
479: [272] ox_execute_string(0," deRham ");
480: 0
481: [273] ox_pop_cmo(0);
482: [1,2]
483: @end example
484: */
485: /*&eg-texi
486: @table @t
487: @item Reference
488: @code{ox_push_cmo}
489: @end table
490: */
491: /*&jp-texi
492: @table @t
493: @item Reference
494: @code{ox_push_cmo}
495: @end table
496: */
497:
498:
499:
500: /*&eg-texi
1.8 takayama 501: @c sort-sm1.gb
502: @node sm1.gb,,, SM1 Functions
503: @subsection @code{sm1.gb}
504: @findex sm1.gb
505: @findex sm1.gb_d
1.1 takayama 506: @table @t
1.8 takayama 507: @item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
1.1 takayama 508: :: computes the Grobner basis of @var{f} in the ring of differential
509: operators with the variable @var{v}.
1.8 takayama 510: @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 511: :: computes the Grobner basis of @var{f} in the ring of differential
512: operators with the variable @var{v}.
513: The result will be returned as a list of distributed polynomials.
514: @end table
515:
516: @table @var
517: @item return
518: List
1.3 takayama 519: @item p, q, r
1.1 takayama 520: Number
521: @item f, v, w
522: List
523: @end table
524:
525: @itemize @bullet
526: @item
527: It returns the Grobner basis of the set of polynomials @var{f}
528: in the ring of deferential operators with the variables @var{v}.
529: @item
530: The weight vectors are given by @var{w}, which can be omitted.
531: If @var{w} is not given,
532: the graded reverse lexicographic order will be used to compute Grobner basis.
533: @item
1.8 takayama 534: The return value of @code{sm1.gb}
1.1 takayama 535: is the list of the Grobner basis of @var{f} and the initial
536: terms (when @var{w} is not given) or initial ideal (when @var{w} is given).
537: @item
1.8 takayama 538: @code{sm1.gb_d} returns the results by a list of distributed polynomials.
1.1 takayama 539: Monomials in each distributed polynomial are ordered in the given order.
540: The return value consists of
541: [variable names, order matrix, grobner basis in districuted polynomials,
542: initial monomials or initial polynomials].
543: @item
544: When a non-term order is given, the Grobner basis is computed in
545: the homogenized Weyl algebra (See Section 1.2 of the book of SST).
546: The homogenization variable h is automatically added.
1.2 takayama 547: @item
1.8 takayama 548: When the optional variable @var{q} is set, @code{sm1.gb} returns,
1.2 takayama 549: as the third return value, a list of
550: the Grobner basis and the initial ideal
551: with sums of monomials sorted by the given order.
552: Each polynomial is expressed as a string temporally for now.
1.3 takayama 553: When the optional variable @var{r} is set to one,
554: the polynomials are dehomogenized (,i.e., h is set to 1).
1.1 takayama 555: @end itemize
556: */
557: /*&jp-texi
1.8 takayama 558: @c sort-sm1.gb
559: @node sm1.gb,,, SM1 Functions
560: @subsection @code{sm1.gb}
561: @findex sm1.gb
562: @findex sm1.gb_d
1.1 takayama 563: @table @t
1.8 takayama 564: @item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
1.1 takayama 565: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
1.8 takayama 566: @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 567: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. $B7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.
568: @end table
569:
570: @table @var
571: @item return
572: $B%j%9%H(B
1.3 takayama 573: @item p, q, r
1.1 takayama 574: $B?t(B
575: @item f, v, w
576: $B%j%9%H(B
577: @end table
578:
579: @itemize @bullet
580: @item
581: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
582: @item
583: Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
584: $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
585: $B%V%l%V%J4pDl$r7W;;$9$k(B.
586: @item
1.8 takayama 587: @code{sm1.gb} $B$NLa$jCM$O(B @var{f} $B$N%0%l%V%J4pDl$*$h$S%$%K%7%c%k%b%N%_%"%k(B
1.1 takayama 588: ( @var{w} $B$,$J$$$H$-(B ) $B$^$?$O(B $B%$%K%7%!%kB?9`<0(B ( @var{w} $B$,M?$($i$?$H$-(B)
589: $B$N%j%9%H$G$"$k(B.
590: @item
1.8 takayama 591: @code{sm1.gb_d} $B$O7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.
1.1 takayama 592: $BB?9`<0$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$$$k(B.
593: $BLa$jCM$O(B
594: [$BJQ?tL>$N%j%9%H(B, $B=g=x$r$-$a$k9TNs(B, $B%0%l%V%J4pDl(B, $B%$%K%7%c%k%b%N%_%"%k$^$?$O%$%K%7%!%kB?9`<0(B]
595: $B$G$"$k(B.
596: @item
597: Term order $B$G$J$$=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).
598: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
1.2 takayama 599: @item $B%*%W%7%g%J%kJQ?t(B @var{q} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
600: 3 $BHVL\$NLa$jCM$H$7$F(B, $B%0%l%V%J4pDl$*$h$S%$%K%7%!%k$N%j%9%H$,(B
601: $BM?$($i$l$?=g=x$G%=!<%H$5$l$?%b%N%_%"%k$NOB$H$7$FLa$5$l$k(B.
602: $B$$$^$N$H$3$m$3$NB?9`<0$O(B, $BJ8;zNs$GI=8=$5$l$k(B.
1.3 takayama 603: $B%*%W%7%g%J%kJQ?t(B @var{r} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
604: $BLa$jB?9`<0$O(B dehomogenize $B$5$l$k(B ($B$9$J$o$A(B h $B$K(B 1 $B$,BeF~$5$l$k(B).
1.1 takayama 605: @end itemize
606: */
607: /*&C-texi
608: @example
1.8 takayama 609: [293] sm1.gb([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
1.1 takayama 610: [[x*dx+y*dy-1,y^2*dy^2+2],[x*dx,y^2*dy^2]]
611: @end example
612: */
613: /*&eg-texi
614: In the example above,
615: @tex the set $\{ x \partial_x + y \partial_y -1,
616: y^2 \partial_y^2+2\}$
617: is the Gr\"obner basis of the input with respect to the
618: graded reverse lexicographic order such that
619: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$.
620: The set $\{x \partial_x, y^2 \partial_y\}$ is the leading monomials
621: (the initial monominals) of the Gr\"obner basis.
622: @end tex
623: */
624: /*&jp-texi
625: $B>e$NNc$K$*$$$F(B,
626: @tex $B=89g(B $\{ x \partial_x + y \partial_y -1,
627: y^2 \partial_y^2+2\}$
628: $B$O(B
629: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$
630: $B$G$"$k$h$&$J(B
631: graded reverse lexicographic order $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
632: $B=89g(B $\{x \partial_x, y^2 \partial_y\}$ $B$O%0%l%V%J4pDl$N3F85$K(B
633: $BBP$9$k(B leading monomial (initial monomial) $B$G$"$k(B.
634: @end tex
635: */
636: /*&C-texi
637: @example
1.8 takayama 638: [294] sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]);
1.1 takayama 639: [[dx+dy^3-4*dy,-dy^4+4*dy^2-1],[dx,-dy^4]]
640: @end example
641: */
642: /*&eg-texi
643: In the example above, two monomials
644: @tex
645: $m = x^a y^b \partial_x^c \partial_y^d$ and
646: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
647: are firstly compared by the weight vector
648: {\tt (dx,dy,x,y) = (50,2,1,0)}
649: (i.e., $m$ is larger than $m'$ if $50c+2d+a > 50c'+2d'+a'$)
650: and when the comparison is tie, then these are
651: compared by the reverse lexicographic order
652: (i.e., if $50c+2d+a = 50c'+2d'+a'$, then use the reverse lexicogrpahic order).
653: @end tex
654: */
655: /*&jp-texi
656: $B>e$NNc$K$*$$$FFs$D$N%b%N%_%"%k(B
657: @tex
658: $m = x^a y^b \partial_x^c \partial_y^d$ $B$*$h$S(B
659: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
660: $B$O:G=i$K(B weight vector
661: {\tt (dx,dy,x,y) = (50,2,1,0)} $B$rMQ$$$FHf3S$5$l$k(B
662: ($B$D$^$j(B $m$ $B$O(B $50c+2d+a > 50c'+2d'+a'$ $B$N$H$-(B
663: $m'$ $B$h$jBg$-$$(B )
664: $B<!$K$3$NHf3S$G>!Ii$,$D$+$J$$$H$-$O(B reverse lexicographic order $B$GHf3S$5$l$k(B
665: ($B$D$^$j(B $50c+2d+a = 50c'+2d'+a'$ $B$N$H$-(B reverse lexicographic order $B$GHf3S(B
666: $B$5$l$k(B).
667: @end tex
1.2 takayama 668: */
669: /*&C-texi
670: @example
1.8 takayama 671: [294] F=sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);
1.2 takayama 672: map(print,F[2][0])$
673: map(print,F[2][1])$
674: @end example
1.1 takayama 675: */
676: /*&C-texi
677: @example
678: [595]
1.8 takayama 679: sm1.gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],
1.1 takayama 680: [x,y],[[dx,1,x,-1],[dy,1]]]);
681:
682: [[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],
683: [x*dx^2+(y*dy-h^2)*dx,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx]]
684:
685: [596]
1.8 takayama 686: sm1.gb_d([["dx (x dx +y dy-2)-1","dy (x dx + y dy -2)-1"],
1.1 takayama 687: "x,y",[[dx,1,x,-1],[dy,1]]]);
688: [[[e0,x,y,H,E,dx,dy,h],
689: [[0,-1,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],
690: [0,1,1,1,1,1,1,0],[0,0,0,0,0,0,-1,0],[0,0,0,0,0,-1,0,0],
691: [0,0,0,0,-1,0,0,0],[0,0,0,-1,0,0,0,0],[0,0,-1,0,0,0,0,0],
692: [0,0,0,0,0,0,0,1]]],
693: [[(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)*
694: <<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
695: ,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
696: ,1,3>>],
697: [(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)*<
698: <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
699: ,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]]
700: @end example
701: */
702:
703: /*&eg-texi
704: @table @t
705: @item Reference
1.8 takayama 706: @code{sm1.reduction}, @code{sm1.rat_to_p}
1.1 takayama 707: @end table
708: */
709: /*&jp-texi
710: @table @t
711: @item $B;2>H(B
1.8 takayama 712: @code{sm1.reduction}, @code{sm1.rat_to_p}
1.1 takayama 713: @end table
714: */
715:
716:
717:
718: /*&eg-texi
1.8 takayama 719: @c sort-sm1.deRham
720: @node sm1.deRham,,, SM1 Functions
721: @subsection @code{sm1.deRham}
722: @findex sm1.deRham
1.1 takayama 723: @table @t
1.8 takayama 724: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
1.1 takayama 725: :: ask the server to evaluate the dimensions of the de Rham cohomology groups
726: of C^n - (the zero set of @var{f}=0).
727: @end table
728:
729: @table @var
730: @item return
731: List
732: @item p
733: Number
734: @item f
735: String or polynomial
736: @item v
737: List
738: @end table
739:
740: @itemize @bullet
741: @item It returns the dimensions of the de Rham cohomology groups
742: of X = C^n \ V(@var{f}).
743: In other words, it returns
744: [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)].
745: @item @var{v} is a list of variables. n = @code{length(@var{v})}.
746: @item
1.8 takayama 747: @code{sm1.deRham} requires huge computer resources.
748: For example, @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
1.1 takayama 749: is already very hard.
750: @item
751: To efficiently analyze the roots of b-function, @code{ox_asir} should be used
752: from @code{ox_sm1_forAsir}.
753: It is recommended to load the communication module for @code{ox_asir}
754: by the command @*
755: @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
756: This command is automatically executed when @code{ox_sm1_forAsir} is started.
1.8 takayama 757: @item If you make an interruption to the function @code{sm1.deRham}
758: by @code{ox_reset(sm1.get_Sm1_proc());}, the server might get out of the standard
1.1 takayama 759: mode. So, it is strongly recommended to execute the command
1.8 takayama 760: @code{ox_shutdown(sm1.get_Sm1_proc());} to interrupt and restart the server.
1.1 takayama 761: @end itemize
762: */
763: /*&jp-texi
1.8 takayama 764: @c sort-sm1.deRham
765: @node sm1.deRham,,, SM1 Functions
766: @subsection @code{sm1.deRham}
767: @findex sm1.deRham
1.1 takayama 768: @table @t
1.8 takayama 769: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
1.1 takayama 770: :: $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.
771: @end table
772:
773: @table @var
774: @item return
775: $B%j%9%H(B
776: @item p
777: $B?t(B
778: @item f
779: $BJ8;zNs(B $B$^$?$O(B $BB?9`<0(B
780: @item v
781: $B%j%9%H(B
782: @end table
783:
784: @itemize @bullet
785: @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.
786: $B$9$J$o$A(B,
787: [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)]
788: $B$rLa$9(B.
789: @item @var{v} $B$OJQ?t$N%j%9%H(B. n = @code{length(@var{v})} $B$G$"$k(B.
790: @item
1.8 takayama 791: @code{sm1.deRham} $B$O7W;;5!$N;q8;$rBgNL$K;HMQ$9$k(B.
792: $B$?$H$($P(B @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
1.1 takayama 793: $B$N7W;;$9$i$9$G$KHs>o$KBgJQ$G$"$k(B.
794: @item
795: b-$B4X?t$N:,$r8zN($h$/2r@O$9$k$K$O(B, @code{ox_asir} $B$,(B @code{ox_sm1_forAsir}
796: $B$h$j;HMQ$5$l$k$Y$-$G$"$k(B. $B%3%^%s%I(B @*
797: @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
798: $B$rMQ$$$F(B, @code{ox_asir} $B$H$NDL?.%b%8%e!<%k$r$"$i$+$8$a%m!<%I$7$F$*$/$H$h$$(B.
799: $B$3$N%3%^%s%I$O(B @code{ox_asir_forAsir} $B$N%9%?!<%H;~$K<+F0E*$K<B9T$5$l$F$$$k(B.
800: @item
1.8 takayama 801: @code{sm1.deRham} $B$r(B @code{ox_reset(sm1.get_Sm1_proc());} $B$GCfCG$9$k$H(B,
1.1 takayama 802: $B0J8e(B sm1 $B%5!<%P$,HsI8=`%b!<%I$KF~$jM=4|$7$J$$F0:n$r$9$k>l9g(B
1.8 takayama 803: $B$,$"$k$N$G(B, $B%3%^%s%I(B @code{ox_shutdown(sm1.get_Sm1_proc());} $B$G(B, @code{ox_sm1_forAsir}
1.1 takayama 804: $B$r0l;~(B shutdown $B$7$F%j%9%?!<%H$7$?J}$,0BA4$G$"$k(B.
805: @end itemize
806: */
807: /*&C-texi
808: @example
1.8 takayama 809: [332] sm1.deRham([x^3-y^2,[x,y]]);
1.1 takayama 810: [1,1,0]
1.8 takayama 811: [333] sm1.deRham([x*(x-1),[x]]);
1.1 takayama 812: [1,2]
813: @end example
814: */
815: /*&eg-texi
816: @table @t
817: @item Reference
1.8 takayama 818: @code{sm1.start}, @code{deRham} (sm1 command)
1.5 takayama 819: @item Algorithm:
1.1 takayama 820: Oaku, Takayama, An algorithm for de Rham cohomology groups of the
821: complement of an affine variety via D-module computation,
822: Journal of pure and applied algebra 139 (1999), 201--233.
823: @end table
824: */
825: /*&jp-texi
826: @table @t
827: @item $B;2>H(B
1.8 takayama 828: @code{sm1.start}, @code{deRham} (sm1 command)
1.5 takayama 829: @item Algorithm:
1.1 takayama 830: Oaku, Takayama, An algorithm for de Rham cohomology groups of the
831: complement of an affine variety via D-module computation,
832: Journal of pure and applied algebra 139 (1999), 201--233.
833: @end table
834: */
835:
836:
837:
838:
839: /*&eg-texi
1.8 takayama 840: @c sort-sm1.hilbert
841: @node sm1.hilbert,,, SM1 Functions
842: @subsection @code{sm1.hilbert}
843: @findex sm1.hilbert
1.1 takayama 844: @findex hilbert_polynomial
845: @table @t
1.8 takayama 846: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
1.1 takayama 847: :: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
848: @item hilbert_polynomial(@var{f},@var{v})
849: :: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
850: @end table
851:
852: @table @var
853: @item return
854: Polynomial
855: @item p
856: Number
857: @item f, v
858: List
859: @end table
860:
861: @itemize @bullet
862: @item It returns the Hilbert polynomial h(k) of the set of polynomials
863: @var{f}
864: with respect to the set of variables @var{v}.
865: @item
866: h(k) = dim_Q F_k/I \cap F_k where F_k the set of polynomials of which
867: degree is less than or equal to k and I is the ideal generated by the
868: set of polynomials @var{f}.
869: @item
1.8 takayama 870: Note for sm1.hilbert:
1.1 takayama 871: For an efficient computation, it is preferable that
872: the set of polynomials @var{f} is a set of monomials.
873: In fact, this function firstly compute a Grobner basis of @var{f}, and then
874: compute the Hilbert polynomial of the initial monomials of the basis.
875: If the input @var{f} is already a Grobner
876: basis, a Grobner basis is recomputed in this function,
877: which is a waste of time and Grobner basis computation in the ring of
878: polynomials in @code{sm1} is slower than in @code{asir}.
879: @end itemize
880: */
881: /*&jp-texi
1.8 takayama 882: @c sort-sm1.hilbert
883: @node sm1.hilbert,,, SM1 Functions
884: @subsection @code{sm1.hilbert}
885: @findex sm1.hilbert
1.1 takayama 886: @findex hilbert_polynomial
887: @table @t
1.8 takayama 888: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
1.1 takayama 889: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
890: @item hilbert_polynomial(@var{f},@var{v})
891: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
892: @end table
893:
894: @table @var
895: @item return
896: $BB?9`<0(B
897: @item p
898: $B?t(B
899: @item f, v
900: $B%j%9%H(B
901: @end table
902:
903: @itemize @bullet
904: @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)
905: $B$r7W;;$9$k(B.
906: @item
907: 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
908: $BB?9`<0$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.
909: @item
1.8 takayama 910: sm1.hilbert $B$K$+$s$9$k%N!<%H(B:
1.1 takayama 911: $B8zN($h$/7W;;$9$k$K$O(B @var{f} $B$O%b%N%_%"%k$N=89g$K$7$?J}$,$$$$(B.
912: $B<B:](B, $B$3$NH!?t$O$^$:(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$7(B, $B$=$l$+$i$=$N(B initial
913: monomial $BC#$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
914: $B$7$?$,$C$F(B, $BF~NO(B @var{f} $B$,$9$G$K%0%l%V%J4pDl$@$H$3$NH!?t$N$J$+$G$b$&0lEY(B
915: $B%0%l%V%J4pDl$N7W;;$,$*$3$J$o$l$k(B. $B$3$l$O;~4V$NL5BL$G$"$k$7(B, @code{sm1} $B$N(B
916: $BB?9`<0%0%l%V%J4pDl7W;;$O(B @code{asir} $B$h$jCY$$(B.
917: @end itemize
918: */
919:
920: /*&C-texi
921: @example
922:
923: [346] load("katsura")$
924: [351] A=hilbert_polynomial(katsura(5),[u0,u1,u2,u3,u4,u5]);
925: 32
926:
927: @end example
928:
929: @example
930: [279] load("katsura")$
931: [280] A=gr(katsura(5),[u0,u1,u2,u3,u4,u5],0)$
932: [281] dp_ord();
933: 0
934: [282] B=map(dp_ht,map(dp_ptod,A,[u0,u1,u2,u3,u4,u5]));
935: [(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>>,
936: (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>>,
937: (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>>,
938: (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>>,
939: (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>>,
940: (1)*<<0,1,0,0,0,4>>,(1)*<<0,0,0,0,0,6>>]
941: [283] C=map(dp_dtop,B,[u0,u1,u2,u3,u4,u5]);
942: [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,
943: 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,
944: u5^4*u2,u5^4*u1,u5^6]
1.8 takayama 945: [284] sm1.hilbert([C,[u0,u1,u2,u3,u4,u5]]);
1.1 takayama 946: 32
947: @end example
948: */
949:
950: /*&eg-texi
951: @table @t
952: @item Reference
1.8 takayama 953: @code{sm1.start}, @code{sm1.gb}, @code{longname}
1.1 takayama 954: @end table
955: */
956: /*&jp-texi
957: @table @t
958: @item $B;2>H(B
1.8 takayama 959: @code{sm1.start}, @code{sm1.gb}, @code{longname}
1.1 takayama 960: @end table
961: */
962:
963:
964: /*&eg-texi
1.8 takayama 965: @c sort-sm1.genericAnn
966: @node sm1.genericAnn,,, SM1 Functions
967: @subsection @code{sm1.genericAnn}
968: @findex sm1.genericAnn
1.1 takayama 969: @table @t
1.8 takayama 970: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
1.1 takayama 971: :: It computes the annihilating ideal for @var{f}^s.
972: @var{v} is the list of variables. Here, s is @var{v}[0] and
973: @var{f} is a polynomial in the variables @code{rest}(@var{v}).
974: @end table
975:
976: @table @var
977: @item return
978: List
979: @item p
980: Number
981: @item f
982: Polynomial
983: @item v
984: List
985: @end table
986:
987: @itemize @bullet
988: @item This function computes the annihilating ideal for @var{f}^s.
989: @var{v} is the list of variables. Here, s is @var{v}[0] and
990: @var{f} is a polynomial in the variables @code{rest}(@var{v}).
991: @end itemize
992: */
993: /*&jp-texi
1.8 takayama 994: @c sort-sm1.genericAnn
995: @node sm1.genericAnn,,, SM1 Functions
996: @subsection @code{sm1.genericAnn}
997: @findex sm1.genericAnn
1.1 takayama 998: @table @t
1.8 takayama 999: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
1.1 takayama 1000: :: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
1001: @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,
1002: @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
1003: @end table
1004:
1005: @table @var
1006: @item return
1007: $B%j%9%H(B
1008: @item p
1009: $B?t(B
1010: @item f
1011: $BB?9`<0(B
1012: @item v
1013: $B%j%9%H(B
1014: @end table
1015:
1016: @itemize @bullet
1017: @item $B$3$NH!?t$O(B,
1018: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
1019: @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,
1020: @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
1021: @end itemize
1022: */
1023: /*&C-texi
1024: @example
1.8 takayama 1025: [595] sm1.genericAnn([x^3+y^3+z^3,[s,x,y,z]]);
1.1 takayama 1026: [-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]
1027: @end example
1028: */
1029: /*&eg-texi
1030: @table @t
1031: @item Reference
1.8 takayama 1032: @code{sm1.start}
1.1 takayama 1033: @end table
1034: */
1035: /*&jp-texi
1036: @table @t
1037: @item $B;2>H(B
1.8 takayama 1038: @code{sm1.start}
1.1 takayama 1039: @end table
1040: */
1041:
1042:
1043:
1044: /*&eg-texi
1.8 takayama 1045: @c sort-sm1.wTensor0
1046: @node sm1.wTensor0,,, SM1 Functions
1047: @subsection @code{sm1.wTensor0}
1048: @findex sm1.wTensor0
1.1 takayama 1049: @table @t
1.8 takayama 1050: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 1051: :: It computes the D-module theoretic 0-th tensor product
1052: of @var{f} and @var{g}.
1053: @end table
1054:
1055: @table @var
1056: @item return
1057: List
1058: @item p
1059: Number
1060: @item f, g, v, w
1061: List
1062: @end table
1063:
1064: @itemize @bullet
1065: @item
1066: It returns the D-module theoretic 0-th tensor product
1067: of @var{f} and @var{g}.
1068: @item
1069: @var{v} is a list of variables.
1070: @var{w} is a list of weights. The integer @var{w}[i] is
1071: the weight of the variable @var{v}[i].
1072: @item
1.8 takayama 1073: @code{sm1.wTensor0} calls @code{wRestriction0} of @code{ox_sm1},
1.1 takayama 1074: which requires a generic weight
1075: vector @var{w} to compute the restriction.
1076: If @var{w} is not generic, the computation fails.
1077: @item Let F and G be solutions of @var{f} and @var{g} respectively.
1078: Intuitively speaking, the 0-th tensor product is a system of
1079: differential equations which annihilates the function FG.
1080: @item The answer is a submodule of a free module D^r in general even if
1081: the inputs @var{f} and @var{g} are left ideals of D.
1082: @end itemize
1083: */
1084:
1085: /*&jp-texi
1.8 takayama 1086: @c sort-sm1.wTensor0
1087: @node sm1.wTensor0,,, SM1 Functions
1088: @subsection @code{sm1.wTensor0}
1089: @findex sm1.wTensor0
1.1 takayama 1090: @table @t
1.8 takayama 1091: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 1092: :: @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
1093: $B7W;;$9$k(B.
1094: @end table
1095:
1096: @table @var
1097: @item return
1098: $B%j%9%H(B
1099: @item p
1100: $B?t(B
1101: @item f, g, v, w
1102: $B%j%9%H(B
1103: @end table
1104:
1105: @itemize @bullet
1106: @item
1107: @var{f} $B$H(B @var{g} $B$N(B
1108: D-$B2C72$H$7$F$N(B 0 $B<!%F%s%=%k@Q$r7W;;$9$k(B.
1109: @item
1110: @var{v} $B$OJQ?t$N%j%9%H$G$"$k(B.
1111: @var{w} $B$O(B weight $B$N%j%9%H$G$"$k(B.
1112: $B@0?t(B @var{w}[i] $B$OJQ?t(B @var{v}[i] $B$N(B weight $B$G$"$k(B.
1113: @item
1.8 takayama 1114: @code{sm1.wTensor0} $B$O(B @code{ox_sm1} $B$N(B @code{wRestriction0}
1.1 takayama 1115: $B$r$h$s$G$$$k(B.
1116: @code{wRestriction0} $B$O(B, generic $B$J(B weight $B%Y%/%H%k(B @var{w}
1117: $B$r$b$H$K$7$F@)8B$r7W;;$7$F$$$k(B.
1118: Weight $B%Y%/%H%k(B @var{w} $B$,(B generic $B$G$J$$$H7W;;$,%(%i!<$GDd;_$9$k(B.
1119: @item F $B$*$h$S(B G $B$r(B @var{f} $B$H(B @var{g} $B$=$l$>$l$N2r$H$9$k(B.
1120: $BD>4QE*$K$$$($P(B, 0 $B<!$N%F%s%=%k@Q$O(B $B4X?t(B FG $B$N$_$?$9HyJ,J}Dx<07O$G$"$k(B.
1121: @item $BF~NO(B @var{f}, @var{g} $B$,(B D $B$N:8%$%G%"%k$G$"$C$F$b(B,
1122: $B0lHL$K(B, $B=PNO$O<+M32C72(B D^r $B$NItJ,2C72$G$"$k(B.
1123: @end itemize
1124: */
1125: /*&C-texi
1126: @example
1.8 takayama 1127: [258] sm1.wTensor0([[x*dx -1, y*dy -4],[dx+dy,dx-dy^2],[x,y],[1,2]]);
1.1 takayama 1128: [[-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],
1129: [-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],
1130: [y^2*dy^2+(-y^2-8*y)*dy+4*y+20]]
1131: @end example
1132: */
1133:
1134:
1135:
1136: /*&eg-texi
1.8 takayama 1137: @c sort-sm1.reduction
1138: @node sm1.reduction,,, SM1 Functions
1139: @subsection @code{sm1.reduction}
1140: @findex sm1.reduction
1.1 takayama 1141: @table @t
1.8 takayama 1142: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 1143: ::
1144: @end table
1145:
1146: @table @var
1147: @item return
1148: List
1149: @item f
1150: Polynomial
1151: @item g, v, w
1152: List
1153: @item p
1154: Number (the process number of ox_sm1)
1155: @end table
1156:
1157: @itemize @bullet
1158: @item It reduces @var{f} by the set of polynomial @var{g}
1159: in the homogenized Weyl algebra; it applies the
1160: division algorithm to @var{f}. The set of variables is @var{v} and
1161: @var{w} is weight vectors to determine the order, which can be ommited.
1.8 takayama 1162: @code{sm1.reduction_noH} is for the Weyl algebra.
1.1 takayama 1163: @item The return value is of the form
1164: [r,c0,[c1,...,cm],[g1,...gm]] where @var{g}=[g1, ..., gm] and
1.7 takayama 1165: c0 f + c1 g1 + ... + cm gm = r.
1.1 takayama 1166: r/c0 is the normal form.
1167: @item The function reduction reduces reducible terms that appear
1168: in lower order terms.
1169: @item The functions
1.8 takayama 1170: sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G)
1.1 takayama 1171: are for distributed polynomials.
1172: @end itemize
1173: */
1174: /*&jp-texi
1.8 takayama 1175: @node sm1.reduction,,, SM1 Functions
1176: @subsection @code{sm1.reduction}
1177: @findex sm1.reduction
1.1 takayama 1178: @table @t
1.8 takayama 1179: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 1180: ::
1181: @end table
1182:
1183: @table @var
1184: @item return
1185: $B%j%9%H(B
1186: @item f
1187: $BB?9`<0(B
1188: @item g, v, w
1189: $B%j%9%H(B
1190: @item p
1191: $B?t(B (ox_sm1 $B$N%W%m%;%9HV9f(B)
1192: @end table
1193:
1194: @itemize @bullet
1195: @item $B$3$NH!?t$O(B @var{f} $B$r(B homogenized $B%o%$%kBe?t$K$*$$$F(B,
1196: $BB?9`<0=89g(B @var{g} $B$G4JC12=(B (reduce) $B$9$k(B; $B$D$^$j(B,
1197: $B$3$NH!?t$O(B, @var{f} $B$K3d;;%"%k%4%j%:%`$rE,MQ$9$k(B.
1198: $BJQ?t=89g$O(B @var{v} $B$G;XDj$9$k(B.
1199: @var{w} $B$O=g=x$r;XDj$9$k$?$a$N(B $B%&%(%$%H%Y%/%H%k$G$"$j(B,
1200: $B>JN,$7$F$b$h$$(B.
1.8 takayama 1201: @code{sm1.reduction_noH} $B$O(B, Weyl algebra $BMQ(B.
1.1 takayama 1202: @item $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
1.7 takayama 1203: [r,c0,[c1,...,cm],g] $B$3$3$G(B @var{g}=[g1, ..., gm] $B$G$"$j(B,
1204: c0 f + c1 g1 + ... + cm gm = r
1.1 takayama 1205: $B$,$J$j$?$D(B.
1206: r/c0 $B$,(B normal form $B$G$"$k(B.
1207: @item $B$3$NH!?t$O(B, $BDc<!9`$K$"$i$o$l$k(B reducible $B$J9`$b4JC12=$9$k(B.
1208: @item $BH!?t(B
1.8 takayama 1209: sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_d(P,F,G)
1.1 takayama 1210: $B$O(B, $BJ,;6B?9`<0MQ$G$"$k(B.
1211: @end itemize
1212: */
1213: /*&C-texi
1214: @example
1.8 takayama 1215: [259] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y]]);
1.7 takayama 1216: [x^2+y^2-4,1,[0,0],[y^4-4*y^2+1,x+y^3-4*y]]
1.8 takayama 1217: [260] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y],[[x,1]]]);
1.7 takayama 1218: [0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]]
1.1 takayama 1219: @end example
1220: */
1221: /*&eg-texi
1222: @table @t
1223: @item Reference
1.10 takayama 1224: @code{sm1.start}, @code{d_true_nf}
1.1 takayama 1225: @end table
1226: */
1227: /*&jp-texi
1228: @table @t
1229: @item $B;2>H(B
1.10 takayama 1230: @code{sm1.start}, @code{d_true_nf}
1.1 takayama 1231: @end table
1232: */
1233:
1234:
1235: /*&eg-texi
1.8 takayama 1236: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
1237: @subsection @code{sm1.xml_tree_to_prefix_string}
1238: @findex sm1.xml_tree_to_prefix_string
1.1 takayama 1239: @table @t
1.8 takayama 1240: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1.1 takayama 1241: :: Translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
1242: @end table
1243:
1244: @table @var
1245: @item return
1246: String
1247: @item p
1248: Number
1249: @item s
1250: String
1251: @end table
1252:
1253: @itemize @bullet
1254: @item It translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
1255: @item This function should be moved to om_* in a future.
1256: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} returns CMO_TREE.
1257: asir has not yet understood this CMO.
1258: @item @code{java} execution environment is required.
1259: (For example, @code{/usr/local/jdk1.1.8/bin} should be in the
1260: command search path.)
1261: @end itemize
1262: */
1263: /*&jp-texi
1.8 takayama 1264: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
1265: @subsection @code{sm1.xml_tree_to_prefix_string}
1266: @findex sm1.xml_tree_to_prefix_string
1.1 takayama 1267: @table @t
1.8 takayama 1268: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1.1 takayama 1269: :: XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
1270: @end table
1271:
1272: @table @var
1273: @item return
1274: String
1275: @item p
1276: Number
1277: @item s
1278: String
1279: @end table
1280:
1281: @itemize @bullet
1282: @item XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
1283: @item $B$3$NH!?t$O(B om_* $B$K>-Mh0\$9$Y$-$G$"$k(B.
1284: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} $B$O(B CMO_TREE
1285: $B$rLa$9(B. @code{asir} $B$O$3$N(B CMO $B$r$^$@%5%]!<%H$7$F$$$J$$(B.
1286: @item @code{java} $B$N<B9T4D6-$,I,MW(B.
1287: ($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.)
1288: @end itemize
1289: */
1290: /*&C-texi
1291: @example
1292: [263] load("om");
1293: 1
1294: [270] F=om_xml(x^4-1);
1295: control: wait OX
1296: Trying to connect to the server... Done.
1297: <OMOBJ><OMA><OMS name="plus" cd="basic"/><OMA>
1298: <OMS name="times" cd="basic"/><OMA>
1299: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>4</OMI></OMA>
1300: <OMI>1</OMI></OMA><OMA><OMS name="times" cd="basic"/><OMA>
1301: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>0</OMI></OMA>
1302: <OMI>-1</OMI></OMA></OMA></OMOBJ>
1.8 takayama 1303: [271] sm1.xml_tree_to_prefix_string(F);
1.1 takayama 1304: basic_plus(basic_times(basic_power(x,4),1),basic_times(basic_power(x,0),-1))
1305: @end example
1306: */
1307: /*&eg-texi
1308: @table @t
1309: @item Reference
1310: @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
1311: @end table
1312: */
1313: /*&jp-texi
1314: @table @t
1315: @item $B;2>H(B
1316: @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
1317: @end table
1318: */
1319:
1320:
1321:
1322:
1323: /*&eg-texi
1.8 takayama 1324: @c sort-sm1.syz
1325: @node sm1.syz,,, SM1 Functions
1326: @subsection @code{sm1.syz}
1327: @findex sm1.syz
1328: @findex sm1.syz_d
1.1 takayama 1329: @table @t
1.8 takayama 1330: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 1331: :: computes the syzygy of @var{f} in the ring of differential
1332: operators with the variable @var{v}.
1333: @end table
1334:
1335: @table @var
1336: @item return
1337: List
1338: @item p
1339: Number
1340: @item f, v, w
1341: List
1342: @end table
1343:
1344: @itemize @bullet
1345: @item
1346: The return values is of the form
1347: [@var{s},[@var{g}, @var{m}, @var{t}]].
1348: Here @var{s} is the syzygy of @var{f} in the ring of differential
1349: operators with the variable @var{v}.
1350: @var{g} is a Groebner basis of @var{f} with the weight vector @var{w},
1351: and @var{m} is a matrix that translates the input matrix @var{f} to the Gr\"obner
1.12 takayama 1352: basis @var{g}.
1.1 takayama 1353: @var{t} is the syzygy of the Gr\"obner basis @var{g}.
1354: In summary, @var{g} = @var{m} @var{f} and
1355: @var{s} @var{f} = 0 hold as matrices.
1356: @item
1357: The weight vectors are given by @var{w}, which can be omitted.
1358: If @var{w} is not given,
1359: the graded reverse lexicographic order will be used to compute Grobner basis.
1360: @item
1361: When a non-term order is given, the Grobner basis is computed in
1362: the homogenized Weyl algebra (See Section 1.2 of the book of SST).
1363: The homogenization variable h is automatically added.
1364: @end itemize
1365: */
1366: /*&jp-texi
1.8 takayama 1367: @c sort-sm1.syz
1368: @node sm1.syz,,, SM1 Functions
1369: @subsection @code{sm1.syz}
1370: @findex sm1.syz
1371: @findex sm1.syz_d
1.1 takayama 1372: @table @t
1.8 takayama 1373: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 1374: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N(B syzygy $B$r7W;;$9$k(B.
1375: @end table
1376:
1377: @table @var
1378: @item return
1379: $B%j%9%H(B
1380: @item p
1381: $B?t(B
1382: @item f, v, w
1383: $B%j%9%H(B
1384: @end table
1385:
1386: @itemize @bullet
1387: @item
1388: $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
1389: [@var{s},[@var{g}, @var{m}, @var{t}]].
1390: $B$3$3$G(B @var{s} $B$O(B @var{f} $B$N(B @var{v} $B$rJQ?t$H$9$kHyJ,:nMQAG4D$K$*$1$k(B
1391: syzygy $B$G$"$k(B.
1392: @var{g} $B$O(B @var{f} $B$N(B weight vector @var{w} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
1393: @var{m} $B$OF~NO9TNs(B @var{f} $B$r%0%l%V%J4pDl(B
1394: @var{g} $B$XJQ49$9$k9TNs$G$"$k(B.
1395: @var{t} $B$O%0%l%V%J4pDl(B @var{g} $B$N(B syzygy $B$G$"$k(B.
1396: $B$^$H$a$k$H(B, $B<!$NEy<0$,$J$j$?$D(B:
1397: @var{g} = @var{m} @var{f} ,
1398: @var{s} @var{f} = 0.
1399: @item
1400: Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
1401: $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
1402: $B%V%l%V%J4pDl$r7W;;$9$k(B.
1403: @item
1404: 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).
1405: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
1406: @end itemize
1407: */
1408: /*&C-texi
1409: @example
1.8 takayama 1410: [293] sm1.syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
1.1 takayama 1411: [[[y*x*dy*dx-2,-x*dx-y*dy+1]], generators of the syzygy
1412: [[[x*dx+y*dy-1],[y^2*dy^2+2]], grobner basis
1413: [[1,0],[y*dy,-1]], transformation matrix
1414: [[y*x*dy*dx-2,-x*dx-y*dy+1]]]]
1415: @end example
1416: */
1417: /*&C-texi
1418: @example
1.8 takayama 1419: [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]]]);
1.1 takayama 1420: [[[y*x*dy*dx-1,-x^2*dx^2-x*dx-y^2*dy^2-y*dy+4]], generators of the syzygy
1421: [[[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
1422: [h^4*x*dx+y^3*dy^3+3*h^2*y^2*dy^2-3*h^4*y*dy]],
1423: [[1,0],[0,1],[y*dy,-x*dx]], transformation matrix
1424: [[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]]]]
1425: @end example
1426: */
1427:
1428:
1429:
1430: /*&eg-texi
1.8 takayama 1431: @node sm1.mul,,, SM1 Functions
1432: @subsection @code{sm1.mul}
1433: @findex sm1.mul
1.1 takayama 1434: @table @t
1.8 takayama 1435: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1.1 takayama 1436: :: ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
1437: @end table
1438:
1439: @table @var
1440: @item return
1441: Polynomial or List
1442: @item p
1443: Number
1444: @item f, g
1445: Polynomial or List
1446: @item v
1447: List
1448: @end table
1449:
1450: @itemize @bullet
1451: @item Ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
1.8 takayama 1452: @item @code{sm1.mul_h} is for homogenized Weyl algebra.
1.1 takayama 1453: @end itemize
1454: */
1455:
1456: /*&jp-texi
1.8 takayama 1457: @node sm1.mul,,, SM1 Functions
1458: @subsection @code{sm1.mul}
1459: @findex sm1.mul
1.1 takayama 1460: @table @t
1.8 takayama 1461: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1.1 takayama 1462: :: sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
1463: $B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
1464: @end table
1465:
1466: @table @var
1467: @item return
1468: $BB?9`<0$^$?$O%j%9%H(B
1469: @item p
1470: $B?t(B
1471: @item f, g
1472: $BB?9`<0$^$?$O%j%9%H(B
1473: @item v
1474: $B%j%9%H(B
1475: @end table
1476:
1477: @itemize @bullet
1478: @item sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
1479: $B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
1.8 takayama 1480: @item @code{sm1.mul_h} $B$O(B homogenized Weyl $BBe?tMQ(B.
1.1 takayama 1481: @end itemize
1482: */
1483:
1484: /*&C-texi
1485:
1486: @example
1.8 takayama 1487: [277] sm1.mul(dx,x,[x]);
1.1 takayama 1488: x*dx+1
1.8 takayama 1489: [278] sm1.mul([x,y],[1,2],[x,y]);
1.1 takayama 1490: x+2*y
1.8 takayama 1491: [279] sm1.mul([[1,2],[3,4]],[[x,y],[1,2]],[x,y]);
1.1 takayama 1492: [[x+2,y+4],[3*x+4,3*y+8]]
1493: @end example
1494:
1495: */
1496:
1497:
1498:
1499:
1500: /*&eg-texi
1.8 takayama 1501: @node sm1.distraction,,, SM1 Functions
1502: @subsection @code{sm1.distraction}
1503: @findex sm1.distraction
1.1 takayama 1504: @table @t
1.8 takayama 1505: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1.1 takayama 1506: :: ask the @code{sm1} server to compute the distraction of @var{f}.
1507: @end table
1508:
1509: @table @var
1510: @item return
1511: List
1512: @item p
1513: Number
1514: @item f
1515: Polynomial
1516: @item v,x,d,s
1517: List
1518: @end table
1519:
1520: @itemize @bullet
1521: @item It asks the @code{sm1} server of the descriptor number @var{p}
1522: to compute the distraction of @var{f} in the ring of differential
1523: operators with variables @var{v}.
1524: @item @var{x} is a list of x-variables and @var{d} is that of d-variables
1525: to be distracted. @var{s} is a list of variables to express the distracted @var{f}.
1526: @item Distraction is roughly speaking to replace x*dx by a single variable x.
1527: See Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations at page 68 for details.
1528: @end itemize
1529: */
1530:
1531: /*&jp-texi
1.8 takayama 1532: @node sm1.distraction,,, SM1 Functions
1.1 takayama 1533:
1.8 takayama 1534: @subsection @code{sm1.distraction}
1535: @findex sm1.distraction
1.1 takayama 1536: @table @t
1.8 takayama 1537: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1.1 takayama 1538: :: @code{sm1} $B$K(B @var{f} $B$N(B distraction $B$r7W;;$7$F$b$i$&(B.
1539: @end table
1540:
1541: @table @var
1542: @item return
1543: $B%j%9%H(B
1544: @item p
1545: $B?t(B
1546: @item f
1547: $BB?9`<0(B
1548: @item v,x,d,s
1549: $B%j%9%H(B
1550: @end table
1551:
1552: @itemize @bullet
1553: @item $B<1JL;R(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B,
1554: @var{f} $B$N(B distraction $B$r(B @var{v} $B>e$NHyJ,:nMQAG4D$G7W;;$7$F$b$i$&(B.
1555: @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
1556: $B%j%9%H(B. Distraction $B$7$?$i(B, @var{s} $B$rJQ?t$H$7$F7k2L$rI=$9(B.
1557: @item Distraction $B$H$$$&$N$O(B x*dx $B$r(B x $B$GCV$-49$($k$3$H$G$"$k(B.
1558: $B>\$7$/$O(B Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations $B$N(B page 68 $B$r8+$h(B.
1559: @end itemize
1560: */
1561:
1562: /*&C-texi
1563:
1564: @example
1.8 takayama 1565: [280] sm1.distraction([x*dx,[x],[x],[dx],[x]]);
1.1 takayama 1566: x
1.8 takayama 1567: [281] sm1.distraction([dx^2,[x],[x],[dx],[x]]);
1.1 takayama 1568: x^2-x
1.8 takayama 1569: [282] sm1.distraction([x^2,[x],[x],[dx],[x]]);
1.1 takayama 1570: x^2+3*x+2
1571: [283] fctr(@@);
1572: [[1,1],[x+1,1],[x+2,1]]
1.8 takayama 1573: [284] sm1.distraction([x*dx*y+x^2*dx^2*dy,[x,y],[x],[dx],[x]]);
1.1 takayama 1574: (x^2-x)*dy+x*y
1575: @end example
1576: */
1577:
1578: /*&eg-texi
1579: @table @t
1580: @item Reference
1581: @code{distraction2(sm1)},
1582: @end table
1583: */
1584:
1585: /*&jp-texi
1586: @table @t
1587: @item $B;2>H(B
1588: @code{distraction2(sm1)},
1589: @end table
1590: */
1591:
1592:
1593:
1594: /*&eg-texi
1.8 takayama 1595: @node sm1.gkz,,, SM1 Functions
1596: @subsection @code{sm1.gkz}
1597: @findex sm1.gkz
1.1 takayama 1598: @table @t
1.8 takayama 1599: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1.1 takayama 1600: :: Returns the GKZ system (A-hypergeometric system) associated to the matrix
1601: @var{A} with the parameter vector @var{B}.
1602: @end table
1603:
1604: @table @var
1605: @item return
1606: List
1607: @item p
1608: Number
1609: @item A, B
1610: List
1611: @end table
1612:
1613: @itemize @bullet
1614: @item Returns the GKZ hypergeometric system
1615: (A-hypergeometric system) associated to the matrix
1616: @end itemize
1617: */
1618:
1619: /*&jp-texi
1.8 takayama 1620: @node sm1.gkz,,, SM1 Functions
1621: @subsection @code{sm1.gkz}
1622: @findex sm1.gkz
1.1 takayama 1623: @table @t
1.8 takayama 1624: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1.1 takayama 1625: :: $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.
1626: @end table
1627:
1628: @table @var
1629: @item return
1630: $B%j%9%H(B
1631: @item p
1632: $B?t(B
1633: @item A, B
1634: $B%j%9%H(B
1635: @end table
1636:
1637: @itemize @bullet
1638: @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.
1639: @end itemize
1640: */
1641:
1642: /*&C-texi
1643:
1644: @example
1645:
1.8 takayama 1646: [280] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
1.1 takayama 1647: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1648: -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1649: [x1,x2,x3,x4]]
1650:
1651: @end example
1652:
1653: */
1654:
1655:
1656:
1657:
1658: /*&eg-texi
1.8 takayama 1659: @node sm1.appell1,,, SM1 Functions
1660: @subsection @code{sm1.appell1}
1661: @findex sm1.appell1
1.1 takayama 1662: @table @t
1.8 takayama 1663: @item sm1.appell1(@var{a}|proc=@var{p})
1.1 takayama 1664: :: Returns the Appell hypergeometric system F_1 or F_D.
1665: @end table
1666:
1667: @table @var
1668: @item return
1669: List
1670: @item p
1671: Number
1672: @item a
1673: List
1674: @end table
1675:
1676: @itemize @bullet
1677: @item Returns the hypergeometric system for the Lauricella function
1678: F_D(a,b1,b2,...,bn,c;x1,...,xn)
1679: where @var{a} =(a,c,b1,...,bn).
1680: When n=2, the Lauricella function is called the Appell function F_1.
1681: The parameters a, c, b1, ..., bn may be rational numbers.
1682: @end itemize
1683: */
1684:
1685: /*&jp-texi
1.8 takayama 1686: @node sm1.appell1,,, SM1 Functions
1687: @subsection @code{sm1.appell1}
1688: @findex sm1.appell1
1.1 takayama 1689: @table @t
1.8 takayama 1690: @item sm1.appell1(@var{a}|proc=@var{p})
1.1 takayama 1691: :: F_1 $B$^$?$O(B F_D $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
1692: @end table
1693:
1694: @table @var
1695: @item return
1696: $B%j%9%H(B
1697: @item p
1698: $B?t(B
1699: @item a
1700: $B%j%9%H(B
1701: @end table
1702:
1703: @itemize @bullet
1704: @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
1705: F_D(a,b1,b2,...,bn,c;x1,...,xn)
1706: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
1707: @var{a} =(a,c,b1,...,bn).
1708: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
1709: @end itemize
1710: */
1711:
1712: /*&C-texi
1713:
1714: @example
1715:
1.8 takayama 1716: [281] sm1.appell1([1,2,3,4]);
1.1 takayama 1717: [[((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2+(-5*x1+2)*dx1-3,
1718: (-x2^2+x2)*dx2^2+((-x1*x2+x1)*dx1-6*x2+2)*dx2-4*x1*dx1-4,
1719: ((-x2+x1)*dx1+3)*dx2-4*dx1], equations
1720: [x1,x2]] the list of variables
1721:
1.8 takayama 1722: [282] sm1.gb(@@);
1.1 takayama 1723: [[((-x2+x1)*dx1+3)*dx2-4*dx1,((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2
1724: +(-5*x1+2)*dx1-3,(-x2^2+x2)*dx2^2+((-x2^2+x1)*dx1-3*x2+2)*dx2
1725: +(-4*x2-4*x1)*dx1-4,
1726: (x2^3+(-x1-1)*x2^2+x1*x2)*dx2^2+((-x1*x2+x1^2)*dx1+6*x2^2
1727: +(-3*x1-2)*x2+2*x1)*dx2-4*x1^2*dx1+4*x2-4*x1],
1728: [x1*dx1*dx2,-x1^2*dx1^2,-x2^2*dx1*dx2,-x1*x2^2*dx2^2]]
1729:
1.8 takayama 1730: [283] sm1.rank(sm1.appell1([1/2,3,5,-1/3]));
1.1 takayama 1731: 1
1732:
1733: [285] Mu=2$ Beta = 1/3$
1.8 takayama 1734: [287] sm1.rank(sm1.appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));
1.1 takayama 1735: 4
1736:
1737:
1738: @end example
1739:
1740: */
1741:
1742: /*&eg-texi
1.8 takayama 1743: @node sm1.appell4,,, SM1 Functions
1744: @subsection @code{sm1.appell4}
1745: @findex sm1.appell4
1.1 takayama 1746: @table @t
1.8 takayama 1747: @item sm1.appell4(@var{a}|proc=@var{p})
1.1 takayama 1748: :: Returns the Appell hypergeometric system F_4 or F_C.
1749: @end table
1750:
1751: @table @var
1752: @item return
1753: List
1754: @item p
1755: Number
1756: @item a
1757: List
1758: @end table
1759:
1760: @itemize @bullet
1761: @item Returns the hypergeometric system for the Lauricella function
1762: F_4(a,b,c1,c2,...,cn;x1,...,xn)
1763: where @var{a} =(a,b,c1,...,cn).
1764: When n=2, the Lauricella function is called the Appell function F_4.
1765: The parameters a, b, c1, ..., cn may be rational numbers.
1766: @end itemize
1767: */
1768:
1769: /*&jp-texi
1.8 takayama 1770: @node sm1.appell4,,, SM1 Functions
1771: @subsection @code{sm1.appell4}
1772: @findex sm1.appell4
1.1 takayama 1773: @table @t
1.8 takayama 1774: @item sm1.appell4(@var{a}|proc=@var{p})
1.1 takayama 1775: :: F_4 $B$^$?$O(B F_C $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
1776: @end table
1777:
1778: @table @var
1779: @item return
1780: $B%j%9%H(B
1781: @item p
1782: $B?t(B
1783: @item a
1784: $B%j%9%H(B
1785: @end table
1786:
1787: @itemize @bullet
1788: @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
1789: F_C(a,b,c1,c2,...,cn;x1,...,xn)
1790: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
1791: @var{a} =(a,b,c1,...,cn).
1792: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
1793: @end itemize
1794: */
1795:
1796: /*&C-texi
1797:
1798: @example
1799:
1.8 takayama 1800: [281] sm1.appell4([1,2,3,4]);
1.1 takayama 1801: [[-x2^2*dx2^2+(-2*x1*x2*dx1-4*x2)*dx2+(-x1^2+x1)*dx1^2+(-4*x1+3)*dx1-2,
1802: (-x2^2+x2)*dx2^2+(-2*x1*x2*dx1-4*x2+4)*dx2-x1^2*dx1^2-4*x1*dx1-2],
1803: equations
1804: [x1,x2]] the list of variables
1805:
1.8 takayama 1806: [282] sm1.rank(@@);
1.1 takayama 1807: 4
1808:
1809: @end example
1810:
1811: */
1812:
1813:
1814:
1815:
1816: /*&eg-texi
1.8 takayama 1817: @node sm1.rank,,, SM1 Functions
1818: @subsection @code{sm1.rank}
1819: @findex sm1.rank
1.1 takayama 1820: @table @t
1.8 takayama 1821: @item sm1.rank(@var{a}|proc=@var{p})
1.1 takayama 1822: :: Returns the holonomic rank of the system of differential equations @var{a}.
1823: @end table
1824:
1825: @table @var
1826: @item return
1827: Number
1828: @item p
1829: Number
1830: @item a
1831: List
1832: @end table
1833:
1834: @itemize @bullet
1835: @item It evaluates the dimension of the space of holomorphic solutions
1836: at a generic point of the system of differential equations @var{a}.
1837: The dimension is called the holonomic rank.
1838: @item @var{a} is a list consisting of a list of differential equations and
1839: a list of variables.
1.8 takayama 1840: @item @code{sm1.rrank} returns the holonomic rank when @var{a} is regular
1841: holonomic. It is generally faster than @code{sm1.rank}.
1.1 takayama 1842: @end itemize
1843: */
1844:
1845: /*&jp-texi
1.8 takayama 1846: @node sm1.rank,,, SM1 Functions
1847: @subsection @code{sm1.rank}
1848: @findex sm1.rank
1.1 takayama 1849: @table @t
1.8 takayama 1850: @item sm1.rank(@var{a}|proc=@var{p})
1.1 takayama 1851: :: $BHyJ,J}Dx<07O(B @var{a} $B$N(B holonomic rank $B$rLa$9(B.
1852: @end table
1853:
1854: @table @var
1855: @item return
1856: $B?t(B
1857: @item p
1858: $B?t(B
1859: @item a
1860: $B%j%9%H(B
1861: @end table
1862:
1863: @itemize @bullet
1864: @item $BHyJ,J}Dx<07O(B @var{a} $B$N(B, generic point $B$G$N@5B'2r$N<!85$r(B
1865: $BLa$9(B. $B$3$N<!85$r(B, holonomic rank $B$H8F$V(B.
1866: @item @var{a} $B$OHyJ,:nMQAG$N%j%9%H$HJQ?t$N%j%9%H$h$j$J$k(B.
1.8 takayama 1867: @item @var{a} $B$,(B regular holonomic $B$N$H$-$O(B @code{sm1.rrank}
1.1 takayama 1868: $B$b(B holonomic rank $B$rLa$9(B.
1.8 takayama 1869: $B$$$C$Q$s$K$3$N4X?t$NJ}$,(B @code{sm1.rank} $B$h$jAa$$(B.
1.1 takayama 1870: @end itemize
1871: */
1872:
1873: /*&C-texi
1874:
1875: @example
1876:
1.8 takayama 1877: [284] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
1.1 takayama 1878: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1879: -dx1*dx4+dx2*dx3, -dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1880: [x1,x2,x3,x4]]
1.8 takayama 1881: [285] sm1.rrank(@@);
1.1 takayama 1882: 4
1883:
1.8 takayama 1884: [286] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [1,2]]);
1.1 takayama 1885: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1886: -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1887: [x1,x2,x3,x4]]
1.8 takayama 1888: [287] sm1.rrank(@@);
1.1 takayama 1889: 5
1890:
1891: @end example
1892:
1893: */
1894:
1895:
1896: /*&eg-texi
1.8 takayama 1897: @node sm1.auto_reduce,,, SM1 Functions
1898: @subsection @code{sm1.auto_reduce}
1899: @findex sm1.auto_reduce
1.1 takayama 1900: @table @t
1.8 takayama 1901: @item sm1.auto_reduce(@var{s}|proc=@var{p})
1.1 takayama 1902: :: Set the flag "AutoReduce" to @var{s}.
1903: @end table
1904:
1905: @table @var
1906: @item return
1907: Number
1908: @item p
1909: Number
1910: @item s
1911: Number
1912: @end table
1913:
1914: @itemize @bullet
1915: @item If @var{s} is 1, then all Grobner bases to be computed
1916: will be the reduced Grobner bases.
1917: @item If @var{s} is 0, then Grobner bases are not necessarily the reduced
1918: Grobner bases. This is the default.
1919: @end itemize
1920: */
1921:
1922: /*&jp-texi
1.8 takayama 1923: @node sm1.auto_reduce,,, SM1 Functions
1924: @subsection @code{sm1.auto_reduce}
1925: @findex sm1.auto_reduce
1.1 takayama 1926: @table @t
1.8 takayama 1927: @item sm1.auto_reduce(@var{s}|proc=@var{p})
1.1 takayama 1928: :: $B%U%i%0(B "AutoReduce" $B$r(B @var{s} $B$K@_Dj(B.
1929: @end table
1930:
1931: @table @var
1932: @item $BLa$jCM(B
1933: $B?t(B
1934: @item p
1935: $B?t(B
1936: @item s
1937: $B?t(B
1938: @end table
1939:
1940: @itemize @bullet
1941: @item @var{s} $B$,(B 1 $B$N$H$-(B, $B0J8e7W;;$5$l$k%0%l%V%J4pDl$O$9$Y$F(B,
1942: reduced $B%0%l%V%J4pDl$H$J$k(B.
1943: @item @var{s} $B$,(B 0 $B$N$H$-(B, $B7W;;$5$l$k%0%l%V%J4pDl$O(B
1944: reduced $B%0%l%V%J4pDl$H$O$+$.$i$J$$(B. $B$3$A$i$,%G%U%)!<%k%H(B.
1945: @end itemize
1946: */
1947:
1948:
1949:
1950: /*&eg-texi
1.8 takayama 1951: @node sm1.slope,,, SM1 Functions
1952: @subsection @code{sm1.slope}
1953: @findex sm1.slope
1.1 takayama 1954: @table @t
1.8 takayama 1955: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
1.1 takayama 1956: :: Returns the slopes of differential equations @var{ii}.
1957: @end table
1958:
1959: @table @var
1960: @item return
1961: List
1962: @item p
1963: Number
1964: @item ii
1965: List (equations)
1966: @item v
1967: List (variables)
1968: @item f_filtration
1969: List (weight vector)
1970: @item v_filtration
1971: List (weight vector)
1972: @end table
1973:
1974: @itemize @bullet
1.8 takayama 1975: @item @code{sm1.slope} returns the (geometric) slopes
1.1 takayama 1976: of the system of differential equations @var{ii}
1977: along the hyperplane specified by
1978: the V filtration @var{v_filtration}.
1979: @item @var{v} is a list of variables.
1980: @item The return value is a list of lists.
1981: The first entry of each list is the slope and the second entry
1982: is the weight vector for which the microcharacteristic variety is
1983: not bihomogeneous.
1984: @end itemize
1.5 takayama 1985:
1986: @noindent
1987: Algorithm:
1988: see "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
1989: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
1990: Note that the signs of the slopes are negative, but the absolute values
1991: of the slopes are returned.
1992:
1.1 takayama 1993: */
1994:
1995: /*&jp-texi
1.8 takayama 1996: @node sm1.slope,,, SM1 Functions
1997: @subsection @code{sm1.slope}
1998: @findex sm1.slope
1.1 takayama 1999: @table @t
1.8 takayama 2000: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
1.1 takayama 2001: :: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B slope $B$rLa$9(B.
2002: @end table
2003:
2004: @table @var
2005: @item return
2006: $B?t(B
2007: @item p
2008: $B?t(B
2009: @item ii
2010: $B%j%9%H(B ($BJ}Dx<0(B)
2011: @item v
2012: $B%j%9%H(B ($BJQ?t(B)
2013: @item f_filtration
2014: $B%j%9%H(B (weight vector)
2015: @item v_filtration
2016: $B%j%9%H(B (weight vector)
2017: @end table
2018:
2019: @itemize @bullet
1.8 takayama 2020: @item @code{sm1.slope} $B$O(B
1.1 takayama 2021: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B V filtration @var{v_filtration}
2022: $B$G;XDj$9$kD6J?LL$K1h$C$F$N(B (geomeric) slope $B$r7W;;$9$k(B.
2023: @item @var{v} $B$OJQ?t$N%j%9%H(B.
1.5 takayama 2024: @item $BLa$jCM$O(B, $B%j%9%H$r@.J,$H$9$k%j%9%H$G$"$k(B.
2025: $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
2026: microcharacteristic variety $B$,(B bihomogeneous $B$G$J$$(B.
2027: @end itemize
2028:
2029: @noindent
2030: Algorithm:
1.1 takayama 2031: "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
2032: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
2033: $B$r$_$h(B.
2034: Slope $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B, $B$3$N%W%m%0%i%`$O(B,
2035: Slope $B$N@dBPCM$rLa$9(B.
2036: */
2037:
2038: /*&C-texi
2039:
2040: @example
2041:
1.8 takayama 2042: [284] A= sm1.gkz([ [[1,2,3]], [-3] ]);
1.1 takayama 2043:
2044:
1.8 takayama 2045: [285] sm1.slope(A[0],A[1],[0,0,0,1,1,1],[0,0,-1,0,0,1]);
1.1 takayama 2046:
1.8 takayama 2047: [286] A2 = sm1.gkz([ [[1,1,1,0],[2,-3,1,-3]], [1,0]]);
1.1 takayama 2048: (* This is an interesting example given by Laura Matusevich,
2049: June 9, 2001 *)
2050:
1.8 takayama 2051: [287] sm1.slope(A2[0],A2[1],[0,0,0,0,1,1,1,1],[0,0,0,-1,0,0,0,1]);
1.1 takayama 2052:
2053:
2054: @end example
2055:
2056: */
2057: /*&eg-texi
2058: @table @t
2059: @item Reference
1.10 takayama 2060: @code{sm.gb}
1.1 takayama 2061: @end table
2062: */
2063: /*&jp-texi
2064: @table @t
2065: @item $B;2>H(B
1.10 takayama 2066: @code{sm.gb}
1.1 takayama 2067: @end table
1.4 takayama 2068: */
2069:
2070:
2071: /*&eg-texi
2072: @include sm1-auto-en.texi
2073: */
2074:
2075: /*&jp-texi
2076: @include sm1-auto-ja.texi
1.1 takayama 2077: */
2078:
2079:
2080: end$
2081:
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