Annotation of OpenXM/src/asir-contrib/packages/doc/sm1.oxweave, Revision 1.10
1.10 ! takayama 1: /*$OpenXM: OpenXM/src/asir-contrib/packages/doc/sm1.oxweave,v 1.9 2003/05/19 05:15:52 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
8: */
1.1 takayama 9: /*&jp-texi
10: @chapter SM1 $BH!?t(B
11:
12: $B$3$N@a$G$O(B sm1 $B$N(B ox $B%5!<%P(B @code{ox_sm1_forAsir}
13: $B$H$N%$%s%?%U%'!<%94X?t$r2r@b$9$k(B.
14: $B$3$l$i$N4X?t$O%U%!%$%k(B @file{sm1} $B$GDj5A$5$l$F$$$k(B.
15: @file{sm1} $B$O(B @file{$(OpenXM_HOME)/lib/asir-contrib} $B$K$"$k(B.
16: $B%7%9%F%`(B @code{sm1} $B$OHyJ,:nMQAG4D$G7W;;$9$k$?$a$N%7%9%F%`$G$"$k(B.
17: $B7W;;Be?t4v2?$N$$$m$$$m$JITJQNL$N7W;;$,HyJ,:nMQAG$N7W;;$K5"Ce$9$k(B.
18: @code{sm1} $B$K$D$$$F$NJ8=q$O(B @code{OpenXM/doc/kan96xx} $B$K$"$k(B.
19:
20: $B$H$3$KCG$j$,$J$$$+$.$j$3$N@a$N$9$Y$F$N4X?t$O(B,
21: $BM-M}?t78?t$N<0$rF~NO$H$7$F$&$1$D$1$J$$(B.
22: $B$9$Y$F$NB?9`<0$N78?t$O@0?t$G$J$$$H$$$1$J$$(B.
23:
24: @tex
25: $B6u4V(B
26: $X:={\bf C} \setminus \{ 0, 1 \} = {\bf C} \setminus V(x(x-1))$
27: $B$N%I%i!<%`%3%[%b%m%872C#$N<!85$r7W;;$7$F$_$h$&(B.
28: $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
29: $B$^$o$kFs$D$N%k!<%W$,(B1$B<!85$N%[%b%m%8!<72$N6u4V$r$O$k(B.
30: $B$7$?$,$C$F(B, 1$B<!85%I%i!<%`%3%[%b%m%872$N<!85$O(B $2$ $B$G$"$k(B.
31: @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
32: $B<!85$rEz$($k(B.
33: @end tex
34: */
35: /*&eg-texi
36: @chapter SM1 Functions
37:
38: This chapter describes interface functions for
39: sm1 ox server @code{ox_sm1_forAsir}.
40: These interface functions are defined in the file @file{sm1}.
41: The file @file{sm1} is @*
42: at @file{$(OpenXM_HOME)/lib/asir/contrib-packages}.
43: The system @code{sm1} is a system to compute in the ring of differential
44: operators.
45: Many constructions of invariants
46: in the computational algebraic geometry reduce
47: to constructions in the ring of differential operators.
48: Documents on @code{sm1} are in
49: the directory @code{OpenXM/doc/kan96xx}.
50:
51: All the coefficients of input polynomials should be
52: integers for most functions in this section.
53: Other functions accept rational numbers as inputs
54: and it will be explicitely noted in each explanation
55: of these functions.
56:
57:
58:
59: @tex
60: Let us evaluate the dimensions of the de Rham cohomology groups
61: of
62: $X:={\bf C} \setminus \{ 0, 1 \} = {\bf C} \setminus V(x(x-1))$.
63: The space $X$ is a two punctured plane, so two loops that encircles the
64: points $x=0$ and $x=1$ respectively spans the first homology group.
65: Hence, the dimension of the first de Rham cohomology group is $2$.
66: @code{sm1} answers the dimensions of the 0th and the first
67: cohomology groups.
68: @end tex
69: */
70: /*&C-texi
71: @example
72:
1.5 takayama 73: @include opening.texi
1.1 takayama 74:
1.8 takayama 75: [283] sm1.deRham([x*(x-1),[x]]);
1.1 takayama 76: [1,2]
77: @end example
78: */
79: /*&C-texi
80: @noindent
81: The author of @code{sm1} : Nobuki Takayama, @code{takayama@@math.sci.kobe-u.ac.jp} @*
82: The author of sm1 packages : Toshinori Oaku, @code{oaku@@twcu.ac.jp} @*
83: Reference: [SST] Saito, M., Sturmfels, B., Takayama, N.,
84: Grobner Deformations of Hypergeometric Differential Equations,
85: 1999, Springer.
86: See the appendix.
87: */
1.6 takayama 88:
89: /*
90: @menu
91: * ox_sm1_forAsir::
1.8 takayama 92: * sm1.start::
1.10 ! takayama 93: * sm1.sm1::
1.8 takayama 94: * sm1.push_int0::
95: * sm1.gb::
96: * sm1.deRham::
97: * sm1.hilbert::
1.6 takayama 98: * hilbert_polynomial::
1.8 takayama 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::
1.6 takayama 112: @end menu
113: */
114:
1.1 takayama 115: /*&jp-texi
116: @section @code{ox_sm1_forAsir} $B%5!<%P(B
117: */
118: /*&eg-texi
119: @section @code{ox_sm1_forAsir} Server
120: */
121:
122: /*&eg-texi
123: @node ox_sm1_forAsir,,, Top
124: @subsection @code{ox_sm1_forAsir}
125: @findex ox_sm1_forAsir
126: @table @t
127: @item ox_sm1_forAsir
128: :: @code{sm1} server for @code{asir}.
129: @end table
130: @itemize @bullet
131: @item
132: @code{ox_sm1_forAsir} is the @code{sm1} server started from asir
1.8 takayama 133: by the command @code{sm1.start}.
1.1 takayama 134: In the standard setting, @*
135: @code{ox_sm1_forAsir} =
136: @file{$(OpenXM_HOME)/lib/sm1/bin/ox_sm1}
137: +
138: @file{$(OpenXM_HOME)/lib/sm1/callsm1.sm1} (macro file) @*
139: +
140: @file{$(OpenXM_HOME)/lib/sm1/callsm1b.sm1} (macro file) @*
141: The macro files @file{callsm1.sm1} and @file{callsm1b.sm1}
142: are searched from
143: current directory, @code{$(LOAD_SM1_PATH)},
144: @code{$(OpenXM_HOME)/lib/sm1},
145: @code{/usr/local/lib/sm1}
146: in this order.
147: @item Note for programmers: See the files
148: @file{$(OpenXM_HOME)/src/kxx/oxserver00.c},
149: @file{$(OpenXM_HOME)/src/kxx/sm1stackmachine.c}
150: to build your own server by reading @code{sm1} macros.
151: @end itemize
152: */
153: /*&jp-texi
154: @node ox_sm1_forAsir,,, Top
155: @subsection @code{ox_sm1_forAsir}
156: @findex ox_sm1_forAsir
157: @table @t
158: @item ox_sm1_forAsir
159: :: @code{asir} $B$N$?$a$N(B @code{sm1} $B%5!<%P(B.
160: @end table
161: @itemize @bullet
162: @item
163: $B%5!<%P(B @code{ox_sm1_forAsir} $B$O(B @code{asir} $B$h$j%3%^%s%I(B
1.8 takayama 164: @code{sm1.start} $B$G5/F0$5$l$k(B @code{sm1} $B%5!<%P$G$"$k(B.
1.1 takayama 165:
166: $BI8=`E*@_Dj$G$O(B, @*
167: @code{ox_sm1_forAsir} =
168: @file{$(OpenXM_HOME)/lib/sm1/bin/ox_sm1}
169: +
170: @file{$(OpenXM_HOME)/lib/sm1/callsm1.sm1} (macro file) @*
171: +
172: @file{$(OpenXM_HOME)/lib/sm1/callsm1b.sm1} (macro file) @*
173: $B$G$"$j(B, $B$3$l$i$N%^%/%m%U%!%$%k$O(B, $B0lHL$K$O(B
174: current directory, @code{$(LOAD_SM1_PATH)},
175: @code{$(OpenXM_HOME)/lib/sm1},
176: @code{/usr/local/lib/sm1}
177: $B$N=gHV$G$5$,$5$l$k(B.
178: @item $B%W%m%0%i%^!<$N$?$a$N%N!<%H(B:
179: @code{sm1} $B%^%/%m$rFI$_9~$s$G<+J,FH<+$N%5!<%P$r:n$k$K$O(B
180: $B<!$N%U%!%$%k$b8+$h(B
181: @file{$(OpenXM_HOME)/src/kxx/oxserver00.c},
182: @file{$(OpenXM_HOME)/src/kxx/sm1stackmachine.c}
183: @end itemize
184: */
185:
186:
187: /*&jp-texi
188: @section $BH!?t0lMw(B
189: */
190: /*&eg-texi
191: @section Functions
192: */
193:
194: /*&eg-texi
1.8 takayama 195: @c sort-sm1.start
196: @node sm1.start,,, SM1 Functions
197: @subsection @code{sm1.start}
198: @findex sm1.start
1.1 takayama 199: @table @t
1.8 takayama 200: @item sm1.start()
1.1 takayama 201: :: Start @code{ox_sm1_forAsir} on the localhost.
202: @end table
203:
204: @table @var
205: @item return
206: Integer
207: @end table
208:
209: @itemize @bullet
210: @item Start @code{ox_sm1_forAsir} on the localhost.
211: It returns the descriptor of @code{ox_sm1_forAsir}.
212: @item Set @code{Xm_noX = 1} to start @code{ox_sm1_forAsir}
213: without a debug window.
214: @item You might have to set suitable orders of variable by the command
215: @code{ord}. For example,
216: when you are working in the
217: ring of differential operators on the variable @code{x} and @code{dx}
218: (@code{dx} stands for
219: @tex $\partial/\partial x$
220: @end tex
221: ),
222: @code{sm1} server assumes that
223: the variable @code{dx} is collected to the right and the variable
224: @code{x} is collected to the left in the printed expression.
225: In the example below, you must not use the variable @code{cc}
226: for computation in @code{sm1}.
227: @item The variables from @code{a} to @code{z} except @code{d} and @code{o}
228: and @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},
229: @code{z0}, ..., @code{z20} can be used as variables for ring of
230: differential operators in default. (cf. @code{Sm1_ord_list} in @code{sm1}).
1.8 takayama 231: @item The descriptor is stored in @code{static Sm1_proc}.
232: The descriptor can be obtained by the function
233: @code{sm1.get_Sm1_proc()}.
1.1 takayama 234: @end itemize
235: */
236: /*&jp-texi
1.8 takayama 237: @c sort-sm1.start
238: @node sm1.start,,, SM1 Functions
239: @subsection @code{sm1.start}
240: @findex sm1.start
1.1 takayama 241: @table @t
1.8 takayama 242: @item sm1.start()
1.1 takayama 243: :: localhost $B$G(B @code{ox_sm1_forAsir} $B$r%9%?!<%H$9$k(B.
244: @end table
245:
246: @table @var
247: @item return
248: $B@0?t(B
249: @end table
250:
251: @itemize @bullet
252: @item localhost $B$G(B @code{ox_sm1_forAsir} $B$r%9%?!<%H$9$k(B.
253: $B%5!<%P(B @code{ox_sm1_forAsir} $B$N<1JLHV9f$rLa$9(B.
254: @item @code{Xm_noX = 1} $B$H$*$/$H%5!<%P(B @code{ox_sm1_forAsir} $B$r%G%P%C%0MQ$N(B
255: $B%&%#%s%I%&$J$7$K5/F0$G$-$k(B.
256: @item $B%3%^%s%I(B @code{ord} $B$rMQ$$$FJQ?t=g=x$r@5$7$/@_Dj$7$F$*$/I,MW$,(B
257: $B$"$k(B.
258: $B$?$H$($P(B,
259: $BJQ?t(B @code{x} $B$H(B @code{dx} $B>e$NHyJ,:nMQAG4D(B
260: (@code{dx} $B$O(B
261: @tex $\partial/\partial x$
262: @end tex
263: $B$KBP1~(B)
264: $B$G7W;;$7$F$$$k$H$-(B,
265: @code{sm1} $B%5!<%P$O<0$r0u:~$7$?$H$-(B,
266: $BJQ?t(B @code{dx} $B$O1&B&$K=8$a$lJQ?t(B
267: @code{x} $B$O:8B&$K$"$D$a$i$l$F$$$k$H2>Dj$7$F$$$k(B.
268: $B<!$NNc$G$O(B, $BJQ?t(B @code{cc} $B$r(B @code{sm1} $B$G$N7W;;$N$?$a$KMQ$$$F$O$$$1$J$$(B.
269: @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,
270: $B$=$l$+$i(B, @code{x0}, ..., @code{x20}, @code{y0}, ..., @code{y20},
271: @code{z0}, ..., @code{z20} $B$O(B, $B%G%U%)!<%k%H$GHyJ,:nMQAG4D$NJQ?t$H$7$F(B
272: $B;H$($k(B (cf. @code{Sm1_ord_list} in @code{sm1}).
1.8 takayama 273: @item $B<1JLHV9f$O(B @code{static Sm1_proc} $B$K3JG<$5$l$k(B.
274: $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 275: @end itemize
276: */
277: /*&C-texi
278: @example
279: [260] ord([da,a,db,b]);
280: [da,a,db,b,dx,dy,dz,x,y,z,dt,ds,t,s,u,v,w,
281: ......... omit ..................
282: ]
283: [261] a*da;
284: a*da
285: [262] cc*dcc;
286: dcc*cc
1.8 takayama 287: [263] sm1.mul(da,a,[a]);
1.1 takayama 288: a*da+1
1.8 takayama 289: [264] sm1.mul(a,da,[a]);
1.1 takayama 290: a*da
291: @end example
292: */
293: /*&eg-texi
294: @table @t
295: @item Reference
1.8 takayama 296: @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
1.1 takayama 297: @code{ord}
298: @end table
299: */
300: /*&jp-texi
301: @table @t
302: @item $B;2>H(B
1.8 takayama 303: @code{ox_launch}, @code{sm1.push_int0}, @code{sm1.push_poly0},
1.1 takayama 304: @code{ord}
305: @end table
306: */
307:
308:
309:
310: /*&eg-texi
311: @c sort-sm1
1.10 ! takayama 312: @node sm1.sm1,,, SM1 Functions
! 313: @subsection @code{sm1.sm1}
! 314: @findex sm1.sm1
1.1 takayama 315: @table @t
1.10 ! takayama 316: @item sm1.sm1(@var{p},@var{s})
1.1 takayama 317: :: ask the @code{sm1} server to execute the command string @var{s}.
318: @end table
319:
320: @table @var
321: @item return
322: Void
323: @item p
324: Number
325: @item s
326: String
327: @end table
328:
329: @itemize @bullet
330: @item It asks the @code{sm1} server of the descriptor number @var{p}
331: to execute the command string @var{s}.
1.9 takayama 332: (In the next example, the descriptor number is 0.)
1.1 takayama 333: @end itemize
334: */
335: /*&jp-texi
1.10 ! takayama 336: @node sm1.sm1,,, SM1 Functions
! 337: @subsection @code{sm1.sm1}
! 338: @findex sm1.sm1
1.1 takayama 339: @table @t
1.10 ! takayama 340: @item sm1.sm1(@var{p},@var{s})
1.1 takayama 341: :: $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.
342: @end table
343:
344: @table @var
345: @item return
346: $B$J$7(B
347: @item p
348: $B?t(B
349: @item s
350: $BJ8;zNs(B
351: @end table
352:
353: @itemize @bullet
354: @item $B<1JLHV9f(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B
355: $B%3%^%s%INs(B @var{s} $B$r<B9T$7$F$/$l$k$h$&$KMj$`(B.
1.9 takayama 356: ($B<!$NNc$G$O(B, $B<1JLHV9f(B 0)
1.1 takayama 357: @end itemize
358: */
359: /*&C-texi
360: @example
1.10 ! takayama 361: [261] sm1.sm1(0," ( (x-1)^2 ) . ");
1.1 takayama 362: 0
363: [262] ox_pop_string(0);
364: x^2-2*x+1
1.10 ! takayama 365: [263] sm1.sm1(0," [(x*(x-1)) [(x)]] deRham ");
1.1 takayama 366: 0
367: [264] ox_pop_string(0);
368: [1 , 2]
369: @end example
370: */
1.10 ! takayama 371:
1.1 takayama 372: /*&jp-texi
373: @table @t
374: @item $B;2>H(B
1.8 takayama 375: @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
1.1 takayama 376: @end table
377: */
378: /*&eg-texi
379: @table @t
380: @item Reference
1.8 takayama 381: @code{sm1.start}, @code{ox_push_int0}, @code{sm1.push_poly0}, @code{sm1.get_Sm1_proc()}.
1.1 takayama 382: @end table
383: */
384:
385:
386: /*&eg-texi
1.8 takayama 387: @c sort-sm1.push_int0
388: @node sm1.push_int0,,, SM1 Functions
389: @subsection @code{sm1.push_int0}
390: @findex sm1.push_int0
1.1 takayama 391: @table @t
1.8 takayama 392: @item sm1.push_int0(@var{p},@var{f})
1.1 takayama 393: :: push the object @var{f} to the server with the descriptor number @var{p}.
394: @end table
395:
396: @table @var
397: @item return
398: Void
399: @item p
400: Number
401: @item f
402: Object
403: @end table
404:
405: @itemize @bullet
406: @item When @code{type(@var{f})} is 2 (recursive polynomial),
407: @var{f} is converted to a string (type == 7)
408: and is sent to the server by @code{ox_push_cmo}.
409: @item When @code{type(@var{f})} is 0 (zero),
410: it is translated to the 32 bit integer zero
411: on the server.
412: Note that @code{ox_push_cmo(@var{p},0)} sends @code{CMO_NULL} to the server.
413: In other words, the server does not get the 32 bit integer 0 nor
414: the bignum 0.
415: @item @code{sm1} integers are classfied into the 32 bit integer and
416: the bignum.
417: When @code{type(@var{f})} is 1 (number), it is translated to the
418: 32 bit integer on the server.
419: Note that @code{ox_push_cmo(@var{p},1234)} send the bignum 1234 to the
420: @code{sm1} server.
421: @item In other cases, @code{ox_push_cmo} is called without data conversion.
422: @end itemize
423: */
424: /*&jp-texi
1.8 takayama 425: @c sort-sm1.push_int0
426: @node sm1.push_int0,,, SM1 Functions
427: @subsection @code{sm1.push_int0}
428: @findex sm1.push_int0
1.1 takayama 429: @table @t
1.8 takayama 430: @item sm1.push_int0(@var{p},@var{f})
1.1 takayama 431: :: $B%*%V%8%'%/%H(B @var{f} $B$r<1JL;R(B @var{p} $B$N%5!<%P$XAw$k(B.
432: @end table
433:
434: @table @var
435: @item return
436: $B$J$7(B
437: @item p
438: $B?t(B
439: @item f
440: $B%*%V%8%'%/%H(B
441: @end table
442:
443: @itemize @bullet
444: @item @code{type(@var{f})} $B$,(B 2 ($B:F5"B?9`<0(B) $B$N$H$-(B,
445: @var{f} $B$OJ8;zNs(B (type == 7) $B$KJQ49$5$l$F(B,
446: @code{ox_push_cmo} $B$rMQ$$$F%5!<%P$XAw$i$l$k(B.
447: @item @code{type(@var{f})} $B$,(B 0 (zero) $B$N$H$-$O(B,
448: $B%5!<%P>e$G$O(B, 32 bit $B@0?t$H2r<a$5$l$k(B.
449: $B$J$*(B @code{ox_push_cmo(P,0)} $B$O%5!<%P$KBP$7$F(B @code{CMO_NULL}
450: $B$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.
451: @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.
452: @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
453: $B$*$/$k(B.
454: @code{ox_push_cmo(@var{p},1234)} $B$O(B bignum $B$N(B 1234 $B$r(B
455: @code{sm1} $B%5!<%P$K$*$/$k$3$H$KCm0U$7$h$&(B.
456: @item $B$=$NB>$N>l9g$K$O(B @code{ox_push_cmo} $B$r%G!<%?7?$NJQ49$J$7$K8F$S=P$9(B.
457: @end itemize
458: */
459: /*&C
460: @example
1.8 takayama 461: [219] P=sm1.start();
1.1 takayama 462: 0
1.8 takayama 463: [220] sm1.push_int0(P,x*dx+1);
1.1 takayama 464: 0
465: [221] A=ox_pop_cmo(P);
466: x*dx+1
467: [223] type(A);
468: 7 (string)
469: @end example
470:
471: @example
1.8 takayama 472: [271] sm1.push_int0(0,[x*(x-1),[x]]);
1.1 takayama 473: 0
474: [272] ox_execute_string(0," deRham ");
475: 0
476: [273] ox_pop_cmo(0);
477: [1,2]
478: @end example
479: */
480: /*&eg-texi
481: @table @t
482: @item Reference
483: @code{ox_push_cmo}
484: @end table
485: */
486: /*&jp-texi
487: @table @t
488: @item Reference
489: @code{ox_push_cmo}
490: @end table
491: */
492:
493:
494:
495: /*&eg-texi
1.8 takayama 496: @c sort-sm1.gb
497: @node sm1.gb,,, SM1 Functions
498: @node sm1.gb_d,,, SM1 Functions
499: @subsection @code{sm1.gb}
500: @findex sm1.gb
501: @findex sm1.gb_d
1.1 takayama 502: @table @t
1.8 takayama 503: @item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
1.1 takayama 504: :: computes the Grobner basis of @var{f} in the ring of differential
505: operators with the variable @var{v}.
1.8 takayama 506: @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 507: :: computes the Grobner basis of @var{f} in the ring of differential
508: operators with the variable @var{v}.
509: The result will be returned as a list of distributed polynomials.
510: @end table
511:
512: @table @var
513: @item return
514: List
1.3 takayama 515: @item p, q, r
1.1 takayama 516: Number
517: @item f, v, w
518: List
519: @end table
520:
521: @itemize @bullet
522: @item
523: It returns the Grobner basis of the set of polynomials @var{f}
524: in the ring of deferential operators with the variables @var{v}.
525: @item
526: The weight vectors are given by @var{w}, which can be omitted.
527: If @var{w} is not given,
528: the graded reverse lexicographic order will be used to compute Grobner basis.
529: @item
1.8 takayama 530: The return value of @code{sm1.gb}
1.1 takayama 531: is the list of the Grobner basis of @var{f} and the initial
532: terms (when @var{w} is not given) or initial ideal (when @var{w} is given).
533: @item
1.8 takayama 534: @code{sm1.gb_d} returns the results by a list of distributed polynomials.
1.1 takayama 535: Monomials in each distributed polynomial are ordered in the given order.
536: The return value consists of
537: [variable names, order matrix, grobner basis in districuted polynomials,
538: initial monomials or initial polynomials].
539: @item
540: When a non-term order is given, the Grobner basis is computed in
541: the homogenized Weyl algebra (See Section 1.2 of the book of SST).
542: The homogenization variable h is automatically added.
1.2 takayama 543: @item
1.8 takayama 544: When the optional variable @var{q} is set, @code{sm1.gb} returns,
1.2 takayama 545: as the third return value, a list of
546: the Grobner basis and the initial ideal
547: with sums of monomials sorted by the given order.
548: Each polynomial is expressed as a string temporally for now.
1.3 takayama 549: When the optional variable @var{r} is set to one,
550: the polynomials are dehomogenized (,i.e., h is set to 1).
1.1 takayama 551: @end itemize
552: */
553: /*&jp-texi
1.8 takayama 554: @c sort-sm1.gb
555: @node sm1.gb,,, SM1 Functions
556: @node sm1.gb_d,,, SM1 Functions
557: @subsection @code{sm1.gb}
558: @findex sm1.gb
559: @findex sm1.gb_d
1.1 takayama 560: @table @t
1.8 takayama 561: @item sm1.gb([@var{f},@var{v},@var{w}]|proc=@var{p},sorted=@var{q},dehomogenize=@var{r})
1.1 takayama 562: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
1.8 takayama 563: @item sm1.gb_d([@var{f},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 564: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B. $B7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.
565: @end table
566:
567: @table @var
568: @item return
569: $B%j%9%H(B
1.3 takayama 570: @item p, q, r
1.1 takayama 571: $B?t(B
572: @item f, v, w
573: $B%j%9%H(B
574: @end table
575:
576: @itemize @bullet
577: @item
578: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$9$k(B.
579: @item
580: Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
581: $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
582: $B%V%l%V%J4pDl$r7W;;$9$k(B.
583: @item
1.8 takayama 584: @code{sm1.gb} $B$NLa$jCM$O(B @var{f} $B$N%0%l%V%J4pDl$*$h$S%$%K%7%c%k%b%N%_%"%k(B
1.1 takayama 585: ( @var{w} $B$,$J$$$H$-(B ) $B$^$?$O(B $B%$%K%7%!%kB?9`<0(B ( @var{w} $B$,M?$($i$?$H$-(B)
586: $B$N%j%9%H$G$"$k(B.
587: @item
1.8 takayama 588: @code{sm1.gb_d} $B$O7k2L$rJ,;6B?9`<0$N%j%9%H$GLa$9(B.
1.1 takayama 589: $BB?9`<0$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$$$k(B.
590: $BLa$jCM$O(B
591: [$BJQ?tL>$N%j%9%H(B, $B=g=x$r$-$a$k9TNs(B, $B%0%l%V%J4pDl(B, $B%$%K%7%c%k%b%N%_%"%k$^$?$O%$%K%7%!%kB?9`<0(B]
592: $B$G$"$k(B.
593: @item
594: Term order $B$G$J$$=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).
595: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
1.2 takayama 596: @item $B%*%W%7%g%J%kJQ?t(B @var{q} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
597: 3 $BHVL\$NLa$jCM$H$7$F(B, $B%0%l%V%J4pDl$*$h$S%$%K%7%!%k$N%j%9%H$,(B
598: $BM?$($i$l$?=g=x$G%=!<%H$5$l$?%b%N%_%"%k$NOB$H$7$FLa$5$l$k(B.
599: $B$$$^$N$H$3$m$3$NB?9`<0$O(B, $BJ8;zNs$GI=8=$5$l$k(B.
1.3 takayama 600: $B%*%W%7%g%J%kJQ?t(B @var{r} $B$,%;%C%H$5$l$F$$$k$H$-$O(B,
601: $BLa$jB?9`<0$O(B dehomogenize $B$5$l$k(B ($B$9$J$o$A(B h $B$K(B 1 $B$,BeF~$5$l$k(B).
1.1 takayama 602: @end itemize
603: */
604: /*&C-texi
605: @example
1.8 takayama 606: [293] sm1.gb([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
1.1 takayama 607: [[x*dx+y*dy-1,y^2*dy^2+2],[x*dx,y^2*dy^2]]
608: @end example
609: */
610: /*&eg-texi
611: In the example above,
612: @tex the set $\{ x \partial_x + y \partial_y -1,
613: y^2 \partial_y^2+2\}$
614: is the Gr\"obner basis of the input with respect to the
615: graded reverse lexicographic order such that
616: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$.
617: The set $\{x \partial_x, y^2 \partial_y\}$ is the leading monomials
618: (the initial monominals) of the Gr\"obner basis.
619: @end tex
620: */
621: /*&jp-texi
622: $B>e$NNc$K$*$$$F(B,
623: @tex $B=89g(B $\{ x \partial_x + y \partial_y -1,
624: y^2 \partial_y^2+2\}$
625: $B$O(B
626: $ 1 \leq \partial_y \leq \partial_x \leq y \leq x \leq \cdots$
627: $B$G$"$k$h$&$J(B
628: graded reverse lexicographic order $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
629: $B=89g(B $\{x \partial_x, y^2 \partial_y\}$ $B$O%0%l%V%J4pDl$N3F85$K(B
630: $BBP$9$k(B leading monomial (initial monomial) $B$G$"$k(B.
631: @end tex
632: */
633: /*&C-texi
634: @example
1.8 takayama 635: [294] sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]);
1.1 takayama 636: [[dx+dy^3-4*dy,-dy^4+4*dy^2-1],[dx,-dy^4]]
637: @end example
638: */
639: /*&eg-texi
640: In the example above, two monomials
641: @tex
642: $m = x^a y^b \partial_x^c \partial_y^d$ and
643: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
644: are firstly compared by the weight vector
645: {\tt (dx,dy,x,y) = (50,2,1,0)}
646: (i.e., $m$ is larger than $m'$ if $50c+2d+a > 50c'+2d'+a'$)
647: and when the comparison is tie, then these are
648: compared by the reverse lexicographic order
649: (i.e., if $50c+2d+a = 50c'+2d'+a'$, then use the reverse lexicogrpahic order).
650: @end tex
651: */
652: /*&jp-texi
653: $B>e$NNc$K$*$$$FFs$D$N%b%N%_%"%k(B
654: @tex
655: $m = x^a y^b \partial_x^c \partial_y^d$ $B$*$h$S(B
656: $m' = x^{a'} y^{b'} \partial_x^{c'} \partial_y^{d'}$
657: $B$O:G=i$K(B weight vector
658: {\tt (dx,dy,x,y) = (50,2,1,0)} $B$rMQ$$$FHf3S$5$l$k(B
659: ($B$D$^$j(B $m$ $B$O(B $50c+2d+a > 50c'+2d'+a'$ $B$N$H$-(B
660: $m'$ $B$h$jBg$-$$(B )
661: $B<!$K$3$NHf3S$G>!Ii$,$D$+$J$$$H$-$O(B reverse lexicographic order $B$GHf3S$5$l$k(B
662: ($B$D$^$j(B $50c+2d+a = 50c'+2d'+a'$ $B$N$H$-(B reverse lexicographic order $B$GHf3S(B
663: $B$5$l$k(B).
664: @end tex
1.2 takayama 665: */
666: /*&C-texi
667: @example
1.8 takayama 668: [294] F=sm1.gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);
1.2 takayama 669: map(print,F[2][0])$
670: map(print,F[2][1])$
671: @end example
1.1 takayama 672: */
673: /*&C-texi
674: @example
675: [595]
1.8 takayama 676: sm1.gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],
1.1 takayama 677: [x,y],[[dx,1,x,-1],[dy,1]]]);
678:
679: [[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],
680: [x*dx^2+(y*dy-h^2)*dx,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx]]
681:
682: [596]
1.8 takayama 683: sm1.gb_d([["dx (x dx +y dy-2)-1","dy (x dx + y dy -2)-1"],
1.1 takayama 684: "x,y",[[dx,1,x,-1],[dy,1]]]);
685: [[[e0,x,y,H,E,dx,dy,h],
686: [[0,-1,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],
687: [0,1,1,1,1,1,1,0],[0,0,0,0,0,0,-1,0],[0,0,0,0,0,-1,0,0],
688: [0,0,0,0,-1,0,0,0],[0,0,0,-1,0,0,0,0],[0,0,-1,0,0,0,0,0],
689: [0,0,0,0,0,0,0,1]]],
690: [[(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)*
691: <<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
692: ,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
693: ,1,3>>],
694: [(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)*<
695: <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
696: ,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]]
697: @end example
698: */
699:
700: /*&eg-texi
701: @table @t
702: @item Reference
1.8 takayama 703: @code{sm1.reduction}, @code{sm1.rat_to_p}
1.1 takayama 704: @end table
705: */
706: /*&jp-texi
707: @table @t
708: @item $B;2>H(B
1.8 takayama 709: @code{sm1.reduction}, @code{sm1.rat_to_p}
1.1 takayama 710: @end table
711: */
712:
713:
714:
715: /*&eg-texi
1.8 takayama 716: @c sort-sm1.deRham
717: @node sm1.deRham,,, SM1 Functions
718: @subsection @code{sm1.deRham}
719: @findex sm1.deRham
1.1 takayama 720: @table @t
1.8 takayama 721: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
1.1 takayama 722: :: ask the server to evaluate the dimensions of the de Rham cohomology groups
723: of C^n - (the zero set of @var{f}=0).
724: @end table
725:
726: @table @var
727: @item return
728: List
729: @item p
730: Number
731: @item f
732: String or polynomial
733: @item v
734: List
735: @end table
736:
737: @itemize @bullet
738: @item It returns the dimensions of the de Rham cohomology groups
739: of X = C^n \ V(@var{f}).
740: In other words, it returns
741: [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)].
742: @item @var{v} is a list of variables. n = @code{length(@var{v})}.
743: @item
1.8 takayama 744: @code{sm1.deRham} requires huge computer resources.
745: For example, @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
1.1 takayama 746: is already very hard.
747: @item
748: To efficiently analyze the roots of b-function, @code{ox_asir} should be used
749: from @code{ox_sm1_forAsir}.
750: It is recommended to load the communication module for @code{ox_asir}
751: by the command @*
752: @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
753: This command is automatically executed when @code{ox_sm1_forAsir} is started.
1.8 takayama 754: @item If you make an interruption to the function @code{sm1.deRham}
755: by @code{ox_reset(sm1.get_Sm1_proc());}, the server might get out of the standard
1.1 takayama 756: mode. So, it is strongly recommended to execute the command
1.8 takayama 757: @code{ox_shutdown(sm1.get_Sm1_proc());} to interrupt and restart the server.
1.1 takayama 758: @end itemize
759: */
760: /*&jp-texi
1.8 takayama 761: @c sort-sm1.deRham
762: @node sm1.deRham,,, SM1 Functions
763: @subsection @code{sm1.deRham}
764: @findex sm1.deRham
1.1 takayama 765: @table @t
1.8 takayama 766: @item sm1.deRham([@var{f},@var{v}]|proc=@var{p})
1.1 takayama 767: :: $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.
768: @end table
769:
770: @table @var
771: @item return
772: $B%j%9%H(B
773: @item p
774: $B?t(B
775: @item f
776: $BJ8;zNs(B $B$^$?$O(B $BB?9`<0(B
777: @item v
778: $B%j%9%H(B
779: @end table
780:
781: @itemize @bullet
782: @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.
783: $B$9$J$o$A(B,
784: [dim H^0(X,C), dim H^1(X,C), dim H^2(X,C), ..., dim H^n(X,C)]
785: $B$rLa$9(B.
786: @item @var{v} $B$OJQ?t$N%j%9%H(B. n = @code{length(@var{v})} $B$G$"$k(B.
787: @item
1.8 takayama 788: @code{sm1.deRham} $B$O7W;;5!$N;q8;$rBgNL$K;HMQ$9$k(B.
789: $B$?$H$($P(B @code{sm1.deRham(0,[x*y*z*(x+y+z-1)*(x-y),[x,y,z]])}
1.1 takayama 790: $B$N7W;;$9$i$9$G$KHs>o$KBgJQ$G$"$k(B.
791: @item
792: b-$B4X?t$N:,$r8zN($h$/2r@O$9$k$K$O(B, @code{ox_asir} $B$,(B @code{ox_sm1_forAsir}
793: $B$h$j;HMQ$5$l$k$Y$-$G$"$k(B. $B%3%^%s%I(B @*
794: @code{sm1(0,"[(parse) (oxasir.sm1) pushfile] extension");}
795: $B$rMQ$$$F(B, @code{ox_asir} $B$H$NDL?.%b%8%e!<%k$r$"$i$+$8$a%m!<%I$7$F$*$/$H$h$$(B.
796: $B$3$N%3%^%s%I$O(B @code{ox_asir_forAsir} $B$N%9%?!<%H;~$K<+F0E*$K<B9T$5$l$F$$$k(B.
797: @item
1.8 takayama 798: @code{sm1.deRham} $B$r(B @code{ox_reset(sm1.get_Sm1_proc());} $B$GCfCG$9$k$H(B,
1.1 takayama 799: $B0J8e(B sm1 $B%5!<%P$,HsI8=`%b!<%I$KF~$jM=4|$7$J$$F0:n$r$9$k>l9g(B
1.8 takayama 800: $B$,$"$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 801: $B$r0l;~(B shutdown $B$7$F%j%9%?!<%H$7$?J}$,0BA4$G$"$k(B.
802: @end itemize
803: */
804: /*&C-texi
805: @example
1.8 takayama 806: [332] sm1.deRham([x^3-y^2,[x,y]]);
1.1 takayama 807: [1,1,0]
1.8 takayama 808: [333] sm1.deRham([x*(x-1),[x]]);
1.1 takayama 809: [1,2]
810: @end example
811: */
812: /*&eg-texi
813: @table @t
814: @item Reference
1.8 takayama 815: @code{sm1.start}, @code{deRham} (sm1 command)
1.5 takayama 816: @item Algorithm:
1.1 takayama 817: Oaku, Takayama, An algorithm for de Rham cohomology groups of the
818: complement of an affine variety via D-module computation,
819: Journal of pure and applied algebra 139 (1999), 201--233.
820: @end table
821: */
822: /*&jp-texi
823: @table @t
824: @item $B;2>H(B
1.8 takayama 825: @code{sm1.start}, @code{deRham} (sm1 command)
1.5 takayama 826: @item Algorithm:
1.1 takayama 827: Oaku, Takayama, An algorithm for de Rham cohomology groups of the
828: complement of an affine variety via D-module computation,
829: Journal of pure and applied algebra 139 (1999), 201--233.
830: @end table
831: */
832:
833:
834:
835:
836: /*&eg-texi
1.8 takayama 837: @c sort-sm1.hilbert
838: @node sm1.hilbert,,, SM1 Functions
839: @subsection @code{sm1.hilbert}
840: @findex sm1.hilbert
1.1 takayama 841: @findex hilbert_polynomial
842: @table @t
1.8 takayama 843: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
1.1 takayama 844: :: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
845: @item hilbert_polynomial(@var{f},@var{v})
846: :: ask the server to compute the Hilbert polynomial for the set of polynomials @var{f}.
847: @end table
848:
849: @table @var
850: @item return
851: Polynomial
852: @item p
853: Number
854: @item f, v
855: List
856: @end table
857:
858: @itemize @bullet
859: @item It returns the Hilbert polynomial h(k) of the set of polynomials
860: @var{f}
861: with respect to the set of variables @var{v}.
862: @item
863: h(k) = dim_Q F_k/I \cap F_k where F_k the set of polynomials of which
864: degree is less than or equal to k and I is the ideal generated by the
865: set of polynomials @var{f}.
866: @item
1.8 takayama 867: Note for sm1.hilbert:
1.1 takayama 868: For an efficient computation, it is preferable that
869: the set of polynomials @var{f} is a set of monomials.
870: In fact, this function firstly compute a Grobner basis of @var{f}, and then
871: compute the Hilbert polynomial of the initial monomials of the basis.
872: If the input @var{f} is already a Grobner
873: basis, a Grobner basis is recomputed in this function,
874: which is a waste of time and Grobner basis computation in the ring of
875: polynomials in @code{sm1} is slower than in @code{asir}.
876: @end itemize
877: */
878: /*&jp-texi
1.8 takayama 879: @c sort-sm1.hilbert
880: @node sm1.hilbert,,, SM1 Functions
881: @subsection @code{sm1.hilbert}
882: @findex sm1.hilbert
1.1 takayama 883: @findex hilbert_polynomial
884: @table @t
1.8 takayama 885: @item sm1.hilbert([@var{f},@var{v}]|proc=@var{p})
1.1 takayama 886: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
887: @item hilbert_polynomial(@var{f},@var{v})
888: :: $BB?9`<0$N=89g(B @var{f} $B$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
889: @end table
890:
891: @table @var
892: @item return
893: $BB?9`<0(B
894: @item p
895: $B?t(B
896: @item f, v
897: $B%j%9%H(B
898: @end table
899:
900: @itemize @bullet
901: @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)
902: $B$r7W;;$9$k(B.
903: @item
904: 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
905: $BB?9`<0$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.
906: @item
1.8 takayama 907: sm1.hilbert $B$K$+$s$9$k%N!<%H(B:
1.1 takayama 908: $B8zN($h$/7W;;$9$k$K$O(B @var{f} $B$O%b%N%_%"%k$N=89g$K$7$?J}$,$$$$(B.
909: $B<B:](B, $B$3$NH!?t$O$^$:(B @var{f} $B$N%0%l%V%J4pDl$r7W;;$7(B, $B$=$l$+$i$=$N(B initial
910: monomial $BC#$N%R%k%Y%k%HB?9`<0$r7W;;$9$k(B.
911: $B$7$?$,$C$F(B, $BF~NO(B @var{f} $B$,$9$G$K%0%l%V%J4pDl$@$H$3$NH!?t$N$J$+$G$b$&0lEY(B
912: $B%0%l%V%J4pDl$N7W;;$,$*$3$J$o$l$k(B. $B$3$l$O;~4V$NL5BL$G$"$k$7(B, @code{sm1} $B$N(B
913: $BB?9`<0%0%l%V%J4pDl7W;;$O(B @code{asir} $B$h$jCY$$(B.
914: @end itemize
915: */
916:
917: /*&C-texi
918: @example
919:
920: [346] load("katsura")$
921: [351] A=hilbert_polynomial(katsura(5),[u0,u1,u2,u3,u4,u5]);
922: 32
923:
924: @end example
925:
926: @example
927: [279] load("katsura")$
928: [280] A=gr(katsura(5),[u0,u1,u2,u3,u4,u5],0)$
929: [281] dp_ord();
930: 0
931: [282] B=map(dp_ht,map(dp_ptod,A,[u0,u1,u2,u3,u4,u5]));
932: [(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>>,
933: (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>>,
934: (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>>,
935: (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>>,
936: (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>>,
937: (1)*<<0,1,0,0,0,4>>,(1)*<<0,0,0,0,0,6>>]
938: [283] C=map(dp_dtop,B,[u0,u1,u2,u3,u4,u5]);
939: [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,
940: 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,
941: u5^4*u2,u5^4*u1,u5^6]
1.8 takayama 942: [284] sm1.hilbert([C,[u0,u1,u2,u3,u4,u5]]);
1.1 takayama 943: 32
944: @end example
945: */
946:
947: /*&eg-texi
948: @table @t
949: @item Reference
1.8 takayama 950: @code{sm1.start}, @code{sm1.gb}, @code{longname}
1.1 takayama 951: @end table
952: */
953: /*&jp-texi
954: @table @t
955: @item $B;2>H(B
1.8 takayama 956: @code{sm1.start}, @code{sm1.gb}, @code{longname}
1.1 takayama 957: @end table
958: */
959:
960:
961: /*&eg-texi
1.8 takayama 962: @c sort-sm1.genericAnn
963: @node sm1.genericAnn,,, SM1 Functions
964: @subsection @code{sm1.genericAnn}
965: @findex sm1.genericAnn
1.1 takayama 966: @table @t
1.8 takayama 967: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
1.1 takayama 968: :: It computes the annihilating ideal for @var{f}^s.
969: @var{v} is the list of variables. Here, s is @var{v}[0] and
970: @var{f} is a polynomial in the variables @code{rest}(@var{v}).
971: @end table
972:
973: @table @var
974: @item return
975: List
976: @item p
977: Number
978: @item f
979: Polynomial
980: @item v
981: List
982: @end table
983:
984: @itemize @bullet
985: @item This function computes the annihilating ideal for @var{f}^s.
986: @var{v} is the list of variables. Here, s is @var{v}[0] and
987: @var{f} is a polynomial in the variables @code{rest}(@var{v}).
988: @end itemize
989: */
990: /*&jp-texi
1.8 takayama 991: @c sort-sm1.genericAnn
992: @node sm1.genericAnn,,, SM1 Functions
993: @subsection @code{sm1.genericAnn}
994: @findex sm1.genericAnn
1.1 takayama 995: @table @t
1.8 takayama 996: @item sm1.genericAnn([@var{f},@var{v}]|proc=@var{p})
1.1 takayama 997: :: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
998: @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,
999: @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
1000: @end table
1001:
1002: @table @var
1003: @item return
1004: $B%j%9%H(B
1005: @item p
1006: $B?t(B
1007: @item f
1008: $BB?9`<0(B
1009: @item v
1010: $B%j%9%H(B
1011: @end table
1012:
1013: @itemize @bullet
1014: @item $B$3$NH!?t$O(B,
1015: @var{f}^s $B$N$_$?$9HyJ,J}Dx<0A4BN$r$b$H$a$k(B.
1016: @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,
1017: @var{f} $B$OJQ?t(B @code{rest}(@var{v}) $B>e$NB?9`<0$G$"$k(B.
1018: @end itemize
1019: */
1020: /*&C-texi
1021: @example
1.8 takayama 1022: [595] sm1.genericAnn([x^3+y^3+z^3,[s,x,y,z]]);
1.1 takayama 1023: [-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]
1024: @end example
1025: */
1026: /*&eg-texi
1027: @table @t
1028: @item Reference
1.8 takayama 1029: @code{sm1.start}
1.1 takayama 1030: @end table
1031: */
1032: /*&jp-texi
1033: @table @t
1034: @item $B;2>H(B
1.8 takayama 1035: @code{sm1.start}
1.1 takayama 1036: @end table
1037: */
1038:
1039:
1040:
1041: /*&eg-texi
1.8 takayama 1042: @c sort-sm1.wTensor0
1043: @node sm1.wTensor0,,, SM1 Functions
1044: @subsection @code{sm1.wTensor0}
1045: @findex sm1.wTensor0
1.1 takayama 1046: @table @t
1.8 takayama 1047: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 1048: :: It computes the D-module theoretic 0-th tensor product
1049: of @var{f} and @var{g}.
1050: @end table
1051:
1052: @table @var
1053: @item return
1054: List
1055: @item p
1056: Number
1057: @item f, g, v, w
1058: List
1059: @end table
1060:
1061: @itemize @bullet
1062: @item
1063: It returns the D-module theoretic 0-th tensor product
1064: of @var{f} and @var{g}.
1065: @item
1066: @var{v} is a list of variables.
1067: @var{w} is a list of weights. The integer @var{w}[i] is
1068: the weight of the variable @var{v}[i].
1069: @item
1.8 takayama 1070: @code{sm1.wTensor0} calls @code{wRestriction0} of @code{ox_sm1},
1.1 takayama 1071: which requires a generic weight
1072: vector @var{w} to compute the restriction.
1073: If @var{w} is not generic, the computation fails.
1074: @item Let F and G be solutions of @var{f} and @var{g} respectively.
1075: Intuitively speaking, the 0-th tensor product is a system of
1076: differential equations which annihilates the function FG.
1077: @item The answer is a submodule of a free module D^r in general even if
1078: the inputs @var{f} and @var{g} are left ideals of D.
1079: @end itemize
1080: */
1081:
1082: /*&jp-texi
1.8 takayama 1083: @c sort-sm1.wTensor0
1084: @node sm1.wTensor0,,, SM1 Functions
1085: @subsection @code{sm1.wTensor0}
1086: @findex sm1.wTensor0
1.1 takayama 1087: @table @t
1.8 takayama 1088: @item sm1.wTensor0([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 1089: :: @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
1090: $B7W;;$9$k(B.
1091: @end table
1092:
1093: @table @var
1094: @item return
1095: $B%j%9%H(B
1096: @item p
1097: $B?t(B
1098: @item f, g, v, w
1099: $B%j%9%H(B
1100: @end table
1101:
1102: @itemize @bullet
1103: @item
1104: @var{f} $B$H(B @var{g} $B$N(B
1105: D-$B2C72$H$7$F$N(B 0 $B<!%F%s%=%k@Q$r7W;;$9$k(B.
1106: @item
1107: @var{v} $B$OJQ?t$N%j%9%H$G$"$k(B.
1108: @var{w} $B$O(B weight $B$N%j%9%H$G$"$k(B.
1109: $B@0?t(B @var{w}[i] $B$OJQ?t(B @var{v}[i] $B$N(B weight $B$G$"$k(B.
1110: @item
1.8 takayama 1111: @code{sm1.wTensor0} $B$O(B @code{ox_sm1} $B$N(B @code{wRestriction0}
1.1 takayama 1112: $B$r$h$s$G$$$k(B.
1113: @code{wRestriction0} $B$O(B, generic $B$J(B weight $B%Y%/%H%k(B @var{w}
1114: $B$r$b$H$K$7$F@)8B$r7W;;$7$F$$$k(B.
1115: Weight $B%Y%/%H%k(B @var{w} $B$,(B generic $B$G$J$$$H7W;;$,%(%i!<$GDd;_$9$k(B.
1116: @item F $B$*$h$S(B G $B$r(B @var{f} $B$H(B @var{g} $B$=$l$>$l$N2r$H$9$k(B.
1117: $BD>4QE*$K$$$($P(B, 0 $B<!$N%F%s%=%k@Q$O(B $B4X?t(B FG $B$N$_$?$9HyJ,J}Dx<07O$G$"$k(B.
1118: @item $BF~NO(B @var{f}, @var{g} $B$,(B D $B$N:8%$%G%"%k$G$"$C$F$b(B,
1119: $B0lHL$K(B, $B=PNO$O<+M32C72(B D^r $B$NItJ,2C72$G$"$k(B.
1120: @end itemize
1121: */
1122: /*&C-texi
1123: @example
1.8 takayama 1124: [258] sm1.wTensor0([[x*dx -1, y*dy -4],[dx+dy,dx-dy^2],[x,y],[1,2]]);
1.1 takayama 1125: [[-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],
1126: [-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],
1127: [y^2*dy^2+(-y^2-8*y)*dy+4*y+20]]
1128: @end example
1129: */
1130:
1131:
1132:
1133: /*&eg-texi
1.8 takayama 1134: @c sort-sm1.reduction
1135: @node sm1.reduction,,, SM1 Functions
1136: @subsection @code{sm1.reduction}
1137: @findex sm1.reduction
1.1 takayama 1138: @table @t
1.8 takayama 1139: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 1140: ::
1141: @end table
1142:
1143: @table @var
1144: @item return
1145: List
1146: @item f
1147: Polynomial
1148: @item g, v, w
1149: List
1150: @item p
1151: Number (the process number of ox_sm1)
1152: @end table
1153:
1154: @itemize @bullet
1155: @item It reduces @var{f} by the set of polynomial @var{g}
1156: in the homogenized Weyl algebra; it applies the
1157: division algorithm to @var{f}. The set of variables is @var{v} and
1158: @var{w} is weight vectors to determine the order, which can be ommited.
1.8 takayama 1159: @code{sm1.reduction_noH} is for the Weyl algebra.
1.1 takayama 1160: @item The return value is of the form
1161: [r,c0,[c1,...,cm],[g1,...gm]] where @var{g}=[g1, ..., gm] and
1.7 takayama 1162: c0 f + c1 g1 + ... + cm gm = r.
1.1 takayama 1163: r/c0 is the normal form.
1164: @item The function reduction reduces reducible terms that appear
1165: in lower order terms.
1166: @item The functions
1.8 takayama 1167: sm1.reduction_d(P,F,G) and sm1.reduction_noH_d(P,F,G)
1.1 takayama 1168: are for distributed polynomials.
1169: @end itemize
1170: */
1171: /*&jp-texi
1.8 takayama 1172: @node sm1.reduction,,, SM1 Functions
1173: @subsection @code{sm1.reduction}
1174: @findex sm1.reduction
1.1 takayama 1175: @table @t
1.8 takayama 1176: @item sm1.reduction([@var{f},@var{g},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 1177: ::
1178: @end table
1179:
1180: @table @var
1181: @item return
1182: $B%j%9%H(B
1183: @item f
1184: $BB?9`<0(B
1185: @item g, v, w
1186: $B%j%9%H(B
1187: @item p
1188: $B?t(B (ox_sm1 $B$N%W%m%;%9HV9f(B)
1189: @end table
1190:
1191: @itemize @bullet
1192: @item $B$3$NH!?t$O(B @var{f} $B$r(B homogenized $B%o%$%kBe?t$K$*$$$F(B,
1193: $BB?9`<0=89g(B @var{g} $B$G4JC12=(B (reduce) $B$9$k(B; $B$D$^$j(B,
1194: $B$3$NH!?t$O(B, @var{f} $B$K3d;;%"%k%4%j%:%`$rE,MQ$9$k(B.
1195: $BJQ?t=89g$O(B @var{v} $B$G;XDj$9$k(B.
1196: @var{w} $B$O=g=x$r;XDj$9$k$?$a$N(B $B%&%(%$%H%Y%/%H%k$G$"$j(B,
1197: $B>JN,$7$F$b$h$$(B.
1.8 takayama 1198: @code{sm1.reduction_noH} $B$O(B, Weyl algebra $BMQ(B.
1.1 takayama 1199: @item $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
1.7 takayama 1200: [r,c0,[c1,...,cm],g] $B$3$3$G(B @var{g}=[g1, ..., gm] $B$G$"$j(B,
1201: c0 f + c1 g1 + ... + cm gm = r
1.1 takayama 1202: $B$,$J$j$?$D(B.
1203: r/c0 $B$,(B normal form $B$G$"$k(B.
1204: @item $B$3$NH!?t$O(B, $BDc<!9`$K$"$i$o$l$k(B reducible $B$J9`$b4JC12=$9$k(B.
1205: @item $BH!?t(B
1.8 takayama 1206: sm1.reduction_d(P,F,G) $B$*$h$S(B sm1.reduction_noH_d(P,F,G)
1.1 takayama 1207: $B$O(B, $BJ,;6B?9`<0MQ$G$"$k(B.
1208: @end itemize
1209: */
1210: /*&C-texi
1211: @example
1.8 takayama 1212: [259] sm1.reduction([x^2+y^2-4,[y^4-4*y^2+1,x+y^3-4*y],[x,y]]);
1.7 takayama 1213: [x^2+y^2-4,1,[0,0],[y^4-4*y^2+1,x+y^3-4*y]]
1.8 takayama 1214: [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 1215: [0,1,[-y^2+4,-x+y^3-4*y],[y^4-4*y^2+1,x+y^3-4*y]]
1.1 takayama 1216: @end example
1217: */
1218: /*&eg-texi
1219: @table @t
1220: @item Reference
1.10 ! takayama 1221: @code{sm1.start}, @code{d_true_nf}
1.1 takayama 1222: @end table
1223: */
1224: /*&jp-texi
1225: @table @t
1226: @item $B;2>H(B
1.10 ! takayama 1227: @code{sm1.start}, @code{d_true_nf}
1.1 takayama 1228: @end table
1229: */
1230:
1231:
1232: /*&eg-texi
1.8 takayama 1233: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
1234: @subsection @code{sm1.xml_tree_to_prefix_string}
1235: @findex sm1.xml_tree_to_prefix_string
1.1 takayama 1236: @table @t
1.8 takayama 1237: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1.1 takayama 1238: :: Translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
1239: @end table
1240:
1241: @table @var
1242: @item return
1243: String
1244: @item p
1245: Number
1246: @item s
1247: String
1248: @end table
1249:
1250: @itemize @bullet
1251: @item It translate OpenMath Tree Expression @var{s} in XML to a prefix notation.
1252: @item This function should be moved to om_* in a future.
1253: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} returns CMO_TREE.
1254: asir has not yet understood this CMO.
1255: @item @code{java} execution environment is required.
1256: (For example, @code{/usr/local/jdk1.1.8/bin} should be in the
1257: command search path.)
1258: @end itemize
1259: */
1260: /*&jp-texi
1.8 takayama 1261: @node sm1.xml_tree_to_prefix_string,,, SM1 Functions
1262: @subsection @code{sm1.xml_tree_to_prefix_string}
1263: @findex sm1.xml_tree_to_prefix_string
1.1 takayama 1264: @table @t
1.8 takayama 1265: @item sm1.xml_tree_to_prefix_string(@var{s}|proc=@var{p})
1.1 takayama 1266: :: XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
1267: @end table
1268:
1269: @table @var
1270: @item return
1271: String
1272: @item p
1273: Number
1274: @item s
1275: String
1276: @end table
1277:
1278: @itemize @bullet
1279: @item XML $B$G=q$+$l$?(B OpenMath $B$NLZI=8=(B @var{s} $B$rA0CV5-K!$K$J$*$9(B.
1280: @item $B$3$NH!?t$O(B om_* $B$K>-Mh0\$9$Y$-$G$"$k(B.
1281: @item @code{om_xml_to_cmo(OpenMath Tree Expression)} $B$O(B CMO_TREE
1282: $B$rLa$9(B. @code{asir} $B$O$3$N(B CMO $B$r$^$@%5%]!<%H$7$F$$$J$$(B.
1283: @item @code{java} $B$N<B9T4D6-$,I,MW(B.
1284: ($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.)
1285: @end itemize
1286: */
1287: /*&C-texi
1288: @example
1289: [263] load("om");
1290: 1
1291: [270] F=om_xml(x^4-1);
1292: control: wait OX
1293: Trying to connect to the server... Done.
1294: <OMOBJ><OMA><OMS name="plus" cd="basic"/><OMA>
1295: <OMS name="times" cd="basic"/><OMA>
1296: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>4</OMI></OMA>
1297: <OMI>1</OMI></OMA><OMA><OMS name="times" cd="basic"/><OMA>
1298: <OMS name="power" cd="basic"/><OMV name="x"/><OMI>0</OMI></OMA>
1299: <OMI>-1</OMI></OMA></OMA></OMOBJ>
1.8 takayama 1300: [271] sm1.xml_tree_to_prefix_string(F);
1.1 takayama 1301: basic_plus(basic_times(basic_power(x,4),1),basic_times(basic_power(x,0),-1))
1302: @end example
1303: */
1304: /*&eg-texi
1305: @table @t
1306: @item Reference
1307: @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
1308: @end table
1309: */
1310: /*&jp-texi
1311: @table @t
1312: @item $B;2>H(B
1313: @code{om_*}, @code{OpenXM/src/OpenMath}, @code{eval_str}
1314: @end table
1315: */
1316:
1317:
1318:
1319:
1320: /*&eg-texi
1.8 takayama 1321: @c sort-sm1.syz
1322: @node sm1.syz,,, SM1 Functions
1323: @node sm1.syz_d,,, SM1 Functions
1324: @subsection @code{sm1.syz}
1325: @findex sm1.syz
1326: @findex sm1.syz_d
1.1 takayama 1327: @table @t
1.8 takayama 1328: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 1329: :: computes the syzygy of @var{f} in the ring of differential
1330: operators with the variable @var{v}.
1331: @end table
1332:
1333: @table @var
1334: @item return
1335: List
1336: @item p
1337: Number
1338: @item f, v, w
1339: List
1340: @end table
1341:
1342: @itemize @bullet
1343: @item
1344: The return values is of the form
1345: [@var{s},[@var{g}, @var{m}, @var{t}]].
1346: Here @var{s} is the syzygy of @var{f} in the ring of differential
1347: operators with the variable @var{v}.
1348: @var{g} is a Groebner basis of @var{f} with the weight vector @var{w},
1349: and @var{m} is a matrix that translates the input matrix @var{f} to the Gr\"obner
1350: basis @var {g}.
1351: @var{t} is the syzygy of the Gr\"obner basis @var{g}.
1352: In summary, @var{g} = @var{m} @var{f} and
1353: @var{s} @var{f} = 0 hold as matrices.
1354: @item
1355: The weight vectors are given by @var{w}, which can be omitted.
1356: If @var{w} is not given,
1357: the graded reverse lexicographic order will be used to compute Grobner basis.
1358: @item
1359: When a non-term order is given, the Grobner basis is computed in
1360: the homogenized Weyl algebra (See Section 1.2 of the book of SST).
1361: The homogenization variable h is automatically added.
1362: @end itemize
1363: */
1364: /*&jp-texi
1.8 takayama 1365: @c sort-sm1.syz
1366: @node sm1.syz,,, SM1 Functions
1367: @node sm1.syz_d,,, SM1 Functions
1368: @subsection @code{sm1.syz}
1369: @findex sm1.syz
1370: @findex sm1.syz_d
1.1 takayama 1371: @table @t
1.8 takayama 1372: @item sm1.syz([@var{f},@var{v},@var{w}]|proc=@var{p})
1.1 takayama 1373: :: @var{v} $B>e$NHyJ,:nMQAG4D$K$*$$$F(B @var{f} $B$N(B syzygy $B$r7W;;$9$k(B.
1374: @end table
1375:
1376: @table @var
1377: @item return
1378: $B%j%9%H(B
1379: @item p
1380: $B?t(B
1381: @item f, v, w
1382: $B%j%9%H(B
1383: @end table
1384:
1385: @itemize @bullet
1386: @item
1387: $BLa$jCM$O<!$N7A$r$7$F$$$k(B:
1388: [@var{s},[@var{g}, @var{m}, @var{t}]].
1389: $B$3$3$G(B @var{s} $B$O(B @var{f} $B$N(B @var{v} $B$rJQ?t$H$9$kHyJ,:nMQAG4D$K$*$1$k(B
1390: syzygy $B$G$"$k(B.
1391: @var{g} $B$O(B @var{f} $B$N(B weight vector @var{w} $B$K4X$9$k%0%l%V%J4pDl$G$"$k(B.
1392: @var{m} $B$OF~NO9TNs(B @var{f} $B$r%0%l%V%J4pDl(B
1393: @var{g} $B$XJQ49$9$k9TNs$G$"$k(B.
1394: @var{t} $B$O%0%l%V%J4pDl(B @var{g} $B$N(B syzygy $B$G$"$k(B.
1395: $B$^$H$a$k$H(B, $B<!$NEy<0$,$J$j$?$D(B:
1396: @var{g} = @var{m} @var{f} ,
1397: @var{s} @var{f} = 0.
1398: @item
1399: Weight $B%Y%/%H%k(B @var{w} $B$O>JN,$7$F$h$$(B.
1400: $B>JN,$7$?>l9g(B, graded reverse lexicographic order $B$r$D$+$C$F(B
1401: $B%V%l%V%J4pDl$r7W;;$9$k(B.
1402: @item
1403: 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).
1404: $BF1<!2=JQ?t(B @code{h} $B$,7k2L$K2C$o$k(B.
1405: @end itemize
1406: */
1407: /*&C-texi
1408: @example
1.8 takayama 1409: [293] sm1.syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
1.1 takayama 1410: [[[y*x*dy*dx-2,-x*dx-y*dy+1]], generators of the syzygy
1411: [[[x*dx+y*dy-1],[y^2*dy^2+2]], grobner basis
1412: [[1,0],[y*dy,-1]], transformation matrix
1413: [[y*x*dy*dx-2,-x*dx-y*dy+1]]]]
1414: @end example
1415: */
1416: /*&C-texi
1417: @example
1.8 takayama 1418: [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 1419: [[[y*x*dy*dx-1,-x^2*dx^2-x*dx-y^2*dy^2-y*dy+4]], generators of the syzygy
1420: [[[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
1421: [h^4*x*dx+y^3*dy^3+3*h^2*y^2*dy^2-3*h^4*y*dy]],
1422: [[1,0],[0,1],[y*dy,-x*dx]], transformation matrix
1423: [[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]]]]
1424: @end example
1425: */
1426:
1427:
1428:
1429: /*&eg-texi
1.8 takayama 1430: @node sm1.mul,,, SM1 Functions
1431: @subsection @code{sm1.mul}
1432: @findex sm1.mul
1.1 takayama 1433: @table @t
1.8 takayama 1434: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1.1 takayama 1435: :: ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
1436: @end table
1437:
1438: @table @var
1439: @item return
1440: Polynomial or List
1441: @item p
1442: Number
1443: @item f, g
1444: Polynomial or List
1445: @item v
1446: List
1447: @end table
1448:
1449: @itemize @bullet
1450: @item Ask the sm1 server to multiply @var{f} and @var{g} in the ring of differential operators over @var{v}.
1.8 takayama 1451: @item @code{sm1.mul_h} is for homogenized Weyl algebra.
1.1 takayama 1452: @end itemize
1453: */
1454:
1455: /*&jp-texi
1.8 takayama 1456: @node sm1.mul,,, SM1 Functions
1457: @subsection @code{sm1.mul}
1458: @findex sm1.mul
1.1 takayama 1459: @table @t
1.8 takayama 1460: @item sm1.mul(@var{f},@var{g},@var{v}|proc=@var{p})
1.1 takayama 1461: :: sm1$B%5!<%P(B $B$K(B @var{f} $B$+$1$k(B @var{g} $B$r(B @var{v}
1462: $B>e$NHyJ,:nMQAG4D$G$d$C$F$/$l$k$h$&$KMj$`(B.
1463: @end table
1464:
1465: @table @var
1466: @item return
1467: $BB?9`<0$^$?$O%j%9%H(B
1468: @item p
1469: $B?t(B
1470: @item f, g
1471: $BB?9`<0$^$?$O%j%9%H(B
1472: @item v
1473: $B%j%9%H(B
1474: @end table
1475:
1476: @itemize @bullet
1477: @item 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.
1.8 takayama 1479: @item @code{sm1.mul_h} $B$O(B homogenized Weyl $BBe?tMQ(B.
1.1 takayama 1480: @end itemize
1481: */
1482:
1483: /*&C-texi
1484:
1485: @example
1.8 takayama 1486: [277] sm1.mul(dx,x,[x]);
1.1 takayama 1487: x*dx+1
1.8 takayama 1488: [278] sm1.mul([x,y],[1,2],[x,y]);
1.1 takayama 1489: x+2*y
1.8 takayama 1490: [279] sm1.mul([[1,2],[3,4]],[[x,y],[1,2]],[x,y]);
1.1 takayama 1491: [[x+2,y+4],[3*x+4,3*y+8]]
1492: @end example
1493:
1494: */
1495:
1496:
1497:
1498:
1499: /*&eg-texi
1.8 takayama 1500: @node sm1.distraction,,, SM1 Functions
1501: @subsection @code{sm1.distraction}
1502: @findex sm1.distraction
1.1 takayama 1503: @table @t
1.8 takayama 1504: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1.1 takayama 1505: :: ask the @code{sm1} server to compute the distraction of @var{f}.
1506: @end table
1507:
1508: @table @var
1509: @item return
1510: List
1511: @item p
1512: Number
1513: @item f
1514: Polynomial
1515: @item v,x,d,s
1516: List
1517: @end table
1518:
1519: @itemize @bullet
1520: @item It asks the @code{sm1} server of the descriptor number @var{p}
1521: to compute the distraction of @var{f} in the ring of differential
1522: operators with variables @var{v}.
1523: @item @var{x} is a list of x-variables and @var{d} is that of d-variables
1524: to be distracted. @var{s} is a list of variables to express the distracted @var{f}.
1525: @item Distraction is roughly speaking to replace x*dx by a single variable x.
1526: See Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations at page 68 for details.
1527: @end itemize
1528: */
1529:
1530: /*&jp-texi
1.8 takayama 1531: @node sm1.distraction,,, SM1 Functions
1.1 takayama 1532:
1.8 takayama 1533: @subsection @code{sm1.distraction}
1534: @findex sm1.distraction
1.1 takayama 1535: @table @t
1.8 takayama 1536: @item sm1.distraction([@var{f},@var{v},@var{x},@var{d},@var{s}]|proc=@var{p})
1.1 takayama 1537: :: @code{sm1} $B$K(B @var{f} $B$N(B distraction $B$r7W;;$7$F$b$i$&(B.
1538: @end table
1539:
1540: @table @var
1541: @item return
1542: $B%j%9%H(B
1543: @item p
1544: $B?t(B
1545: @item f
1546: $BB?9`<0(B
1547: @item v,x,d,s
1548: $B%j%9%H(B
1549: @end table
1550:
1551: @itemize @bullet
1552: @item $B<1JL;R(B @var{p} $B$N(B @code{sm1} $B%5!<%P$K(B,
1553: @var{f} $B$N(B distraction $B$r(B @var{v} $B>e$NHyJ,:nMQAG4D$G7W;;$7$F$b$i$&(B.
1554: @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
1555: $B%j%9%H(B. Distraction $B$7$?$i(B, @var{s} $B$rJQ?t$H$7$F7k2L$rI=$9(B.
1556: @item Distraction $B$H$$$&$N$O(B x*dx $B$r(B x $B$GCV$-49$($k$3$H$G$"$k(B.
1557: $B>\$7$/$O(B Saito, Sturmfels, Takayama : Grobner Deformations of Hypergeometric Differential Equations $B$N(B page 68 $B$r8+$h(B.
1558: @end itemize
1559: */
1560:
1561: /*&C-texi
1562:
1563: @example
1.8 takayama 1564: [280] sm1.distraction([x*dx,[x],[x],[dx],[x]]);
1.1 takayama 1565: x
1.8 takayama 1566: [281] sm1.distraction([dx^2,[x],[x],[dx],[x]]);
1.1 takayama 1567: x^2-x
1.8 takayama 1568: [282] sm1.distraction([x^2,[x],[x],[dx],[x]]);
1.1 takayama 1569: x^2+3*x+2
1570: [283] fctr(@@);
1571: [[1,1],[x+1,1],[x+2,1]]
1.8 takayama 1572: [284] sm1.distraction([x*dx*y+x^2*dx^2*dy,[x,y],[x],[dx],[x]]);
1.1 takayama 1573: (x^2-x)*dy+x*y
1574: @end example
1575: */
1576:
1577: /*&eg-texi
1578: @table @t
1579: @item Reference
1580: @code{distraction2(sm1)},
1581: @end table
1582: */
1583:
1584: /*&jp-texi
1585: @table @t
1586: @item $B;2>H(B
1587: @code{distraction2(sm1)},
1588: @end table
1589: */
1590:
1591:
1592:
1593: /*&eg-texi
1.8 takayama 1594: @node sm1.gkz,,, SM1 Functions
1595: @subsection @code{sm1.gkz}
1596: @findex sm1.gkz
1.1 takayama 1597: @table @t
1.8 takayama 1598: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1.1 takayama 1599: :: Returns the GKZ system (A-hypergeometric system) associated to the matrix
1600: @var{A} with the parameter vector @var{B}.
1601: @end table
1602:
1603: @table @var
1604: @item return
1605: List
1606: @item p
1607: Number
1608: @item A, B
1609: List
1610: @end table
1611:
1612: @itemize @bullet
1613: @item Returns the GKZ hypergeometric system
1614: (A-hypergeometric system) associated to the matrix
1615: @end itemize
1616: */
1617:
1618: /*&jp-texi
1.8 takayama 1619: @node sm1.gkz,,, SM1 Functions
1620: @subsection @code{sm1.gkz}
1621: @findex sm1.gkz
1.1 takayama 1622: @table @t
1.8 takayama 1623: @item sm1.gkz([@var{A},@var{B}]|proc=@var{p})
1.1 takayama 1624: :: $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.
1625: @end table
1626:
1627: @table @var
1628: @item return
1629: $B%j%9%H(B
1630: @item p
1631: $B?t(B
1632: @item A, B
1633: $B%j%9%H(B
1634: @end table
1635:
1636: @itemize @bullet
1637: @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.
1638: @end itemize
1639: */
1640:
1641: /*&C-texi
1642:
1643: @example
1644:
1.8 takayama 1645: [280] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
1.1 takayama 1646: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1647: -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1648: [x1,x2,x3,x4]]
1649:
1650: @end example
1651:
1652: */
1653:
1654:
1655:
1656:
1657: /*&eg-texi
1.8 takayama 1658: @node sm1.appell1,,, SM1 Functions
1659: @subsection @code{sm1.appell1}
1660: @findex sm1.appell1
1.1 takayama 1661: @table @t
1.8 takayama 1662: @item sm1.appell1(@var{a}|proc=@var{p})
1.1 takayama 1663: :: Returns the Appell hypergeometric system F_1 or F_D.
1664: @end table
1665:
1666: @table @var
1667: @item return
1668: List
1669: @item p
1670: Number
1671: @item a
1672: List
1673: @end table
1674:
1675: @itemize @bullet
1676: @item Returns the hypergeometric system for the Lauricella function
1677: F_D(a,b1,b2,...,bn,c;x1,...,xn)
1678: where @var{a} =(a,c,b1,...,bn).
1679: When n=2, the Lauricella function is called the Appell function F_1.
1680: The parameters a, c, b1, ..., bn may be rational numbers.
1681: @end itemize
1682: */
1683:
1684: /*&jp-texi
1.8 takayama 1685: @node sm1.appell1,,, SM1 Functions
1686: @subsection @code{sm1.appell1}
1687: @findex sm1.appell1
1.1 takayama 1688: @table @t
1.8 takayama 1689: @item sm1.appell1(@var{a}|proc=@var{p})
1.1 takayama 1690: :: F_1 $B$^$?$O(B F_D $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
1691: @end table
1692:
1693: @table @var
1694: @item return
1695: $B%j%9%H(B
1696: @item p
1697: $B?t(B
1698: @item a
1699: $B%j%9%H(B
1700: @end table
1701:
1702: @itemize @bullet
1703: @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
1704: F_D(a,b1,b2,...,bn,c;x1,...,xn)
1705: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
1706: @var{a} =(a,c,b1,...,bn).
1707: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
1708: @end itemize
1709: */
1710:
1711: /*&C-texi
1712:
1713: @example
1714:
1.8 takayama 1715: [281] sm1.appell1([1,2,3,4]);
1.1 takayama 1716: [[((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2+(-5*x1+2)*dx1-3,
1717: (-x2^2+x2)*dx2^2+((-x1*x2+x1)*dx1-6*x2+2)*dx2-4*x1*dx1-4,
1718: ((-x2+x1)*dx1+3)*dx2-4*dx1], equations
1719: [x1,x2]] the list of variables
1720:
1.8 takayama 1721: [282] sm1.gb(@@);
1.1 takayama 1722: [[((-x2+x1)*dx1+3)*dx2-4*dx1,((-x1+1)*x2*dx1-3*x2)*dx2+(-x1^2+x1)*dx1^2
1723: +(-5*x1+2)*dx1-3,(-x2^2+x2)*dx2^2+((-x2^2+x1)*dx1-3*x2+2)*dx2
1724: +(-4*x2-4*x1)*dx1-4,
1725: (x2^3+(-x1-1)*x2^2+x1*x2)*dx2^2+((-x1*x2+x1^2)*dx1+6*x2^2
1726: +(-3*x1-2)*x2+2*x1)*dx2-4*x1^2*dx1+4*x2-4*x1],
1727: [x1*dx1*dx2,-x1^2*dx1^2,-x2^2*dx1*dx2,-x1*x2^2*dx2^2]]
1728:
1.8 takayama 1729: [283] sm1.rank(sm1.appell1([1/2,3,5,-1/3]));
1.1 takayama 1730: 1
1731:
1732: [285] Mu=2$ Beta = 1/3$
1.8 takayama 1733: [287] sm1.rank(sm1.appell1([Mu+Beta,Mu+1,Beta,Beta,Beta]));
1.1 takayama 1734: 4
1735:
1736:
1737: @end example
1738:
1739: */
1740:
1741: /*&eg-texi
1.8 takayama 1742: @node sm1.appell4,,, SM1 Functions
1743: @subsection @code{sm1.appell4}
1744: @findex sm1.appell4
1.1 takayama 1745: @table @t
1.8 takayama 1746: @item sm1.appell4(@var{a}|proc=@var{p})
1.1 takayama 1747: :: Returns the Appell hypergeometric system F_4 or F_C.
1748: @end table
1749:
1750: @table @var
1751: @item return
1752: List
1753: @item p
1754: Number
1755: @item a
1756: List
1757: @end table
1758:
1759: @itemize @bullet
1760: @item Returns the hypergeometric system for the Lauricella function
1761: F_4(a,b,c1,c2,...,cn;x1,...,xn)
1762: where @var{a} =(a,b,c1,...,cn).
1763: When n=2, the Lauricella function is called the Appell function F_4.
1764: The parameters a, b, c1, ..., cn may be rational numbers.
1765: @end itemize
1766: */
1767:
1768: /*&jp-texi
1.8 takayama 1769: @node sm1.appell4,,, SM1 Functions
1770: @subsection @code{sm1.appell4}
1771: @findex sm1.appell4
1.1 takayama 1772: @table @t
1.8 takayama 1773: @item sm1.appell4(@var{a}|proc=@var{p})
1.1 takayama 1774: :: F_4 $B$^$?$O(B F_C $B$KBP1~$9$kJ}Dx<07O$rLa$9(B.
1775: @end table
1776:
1777: @table @var
1778: @item return
1779: $B%j%9%H(B
1780: @item p
1781: $B?t(B
1782: @item a
1783: $B%j%9%H(B
1784: @end table
1785:
1786: @itemize @bullet
1787: @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
1788: F_C(a,b,c1,c2,...,cn;x1,...,xn)
1789: $B$N$_$?$9HyJ,J}Dx<07O$rLa$9(B. $B$3$3$G(B,
1790: @var{a} =(a,b,c1,...,cn).
1791: $B%Q%i%a!<%?$OM-M}?t$G$b$h$$(B.
1792: @end itemize
1793: */
1794:
1795: /*&C-texi
1796:
1797: @example
1798:
1.8 takayama 1799: [281] sm1.appell4([1,2,3,4]);
1.1 takayama 1800: [[-x2^2*dx2^2+(-2*x1*x2*dx1-4*x2)*dx2+(-x1^2+x1)*dx1^2+(-4*x1+3)*dx1-2,
1801: (-x2^2+x2)*dx2^2+(-2*x1*x2*dx1-4*x2+4)*dx2-x1^2*dx1^2-4*x1*dx1-2],
1802: equations
1803: [x1,x2]] the list of variables
1804:
1.8 takayama 1805: [282] sm1.rank(@@);
1.1 takayama 1806: 4
1807:
1808: @end example
1809:
1810: */
1811:
1812:
1813:
1814:
1815: /*&eg-texi
1.8 takayama 1816: @node sm1.rank,,, SM1 Functions
1817: @subsection @code{sm1.rank}
1818: @findex sm1.rank
1.1 takayama 1819: @table @t
1.8 takayama 1820: @item sm1.rank(@var{a}|proc=@var{p})
1.1 takayama 1821: :: Returns the holonomic rank of the system of differential equations @var{a}.
1822: @end table
1823:
1824: @table @var
1825: @item return
1826: Number
1827: @item p
1828: Number
1829: @item a
1830: List
1831: @end table
1832:
1833: @itemize @bullet
1834: @item It evaluates the dimension of the space of holomorphic solutions
1835: at a generic point of the system of differential equations @var{a}.
1836: The dimension is called the holonomic rank.
1837: @item @var{a} is a list consisting of a list of differential equations and
1838: a list of variables.
1.8 takayama 1839: @item @code{sm1.rrank} returns the holonomic rank when @var{a} is regular
1840: holonomic. It is generally faster than @code{sm1.rank}.
1.1 takayama 1841: @end itemize
1842: */
1843:
1844: /*&jp-texi
1.8 takayama 1845: @node sm1.rank,,, SM1 Functions
1846: @subsection @code{sm1.rank}
1847: @findex sm1.rank
1.1 takayama 1848: @table @t
1.8 takayama 1849: @item sm1.rank(@var{a}|proc=@var{p})
1.1 takayama 1850: :: $BHyJ,J}Dx<07O(B @var{a} $B$N(B holonomic rank $B$rLa$9(B.
1851: @end table
1852:
1853: @table @var
1854: @item return
1855: $B?t(B
1856: @item p
1857: $B?t(B
1858: @item a
1859: $B%j%9%H(B
1860: @end table
1861:
1862: @itemize @bullet
1863: @item $BHyJ,J}Dx<07O(B @var{a} $B$N(B, generic point $B$G$N@5B'2r$N<!85$r(B
1864: $BLa$9(B. $B$3$N<!85$r(B, holonomic rank $B$H8F$V(B.
1865: @item @var{a} $B$OHyJ,:nMQAG$N%j%9%H$HJQ?t$N%j%9%H$h$j$J$k(B.
1.8 takayama 1866: @item @var{a} $B$,(B regular holonomic $B$N$H$-$O(B @code{sm1.rrank}
1.1 takayama 1867: $B$b(B holonomic rank $B$rLa$9(B.
1.8 takayama 1868: $B$$$C$Q$s$K$3$N4X?t$NJ}$,(B @code{sm1.rank} $B$h$jAa$$(B.
1.1 takayama 1869: @end itemize
1870: */
1871:
1872: /*&C-texi
1873:
1874: @example
1875:
1.8 takayama 1876: [284] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [0,2] ]);
1.1 takayama 1877: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1878: -dx1*dx4+dx2*dx3, -dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1879: [x1,x2,x3,x4]]
1.8 takayama 1880: [285] sm1.rrank(@@);
1.1 takayama 1881: 4
1882:
1.8 takayama 1883: [286] sm1.gkz([ [[1,1,1,1],[0,1,3,4]], [1,2]]);
1.1 takayama 1884: [[x4*dx4+x3*dx3+x2*dx2+x1*dx1-1,4*x4*dx4+3*x3*dx3+x2*dx2-2,
1885: -dx1*dx4+dx2*dx3,-dx2^2*dx4+dx1*dx3^2,dx1^2*dx3-dx2^3,-dx2*dx4^2+dx3^3],
1886: [x1,x2,x3,x4]]
1.8 takayama 1887: [287] sm1.rrank(@@);
1.1 takayama 1888: 5
1889:
1890: @end example
1891:
1892: */
1893:
1894:
1895: /*&eg-texi
1.8 takayama 1896: @node sm1.auto_reduce,,, SM1 Functions
1897: @subsection @code{sm1.auto_reduce}
1898: @findex sm1.auto_reduce
1.1 takayama 1899: @table @t
1.8 takayama 1900: @item sm1.auto_reduce(@var{s}|proc=@var{p})
1.1 takayama 1901: :: Set the flag "AutoReduce" to @var{s}.
1902: @end table
1903:
1904: @table @var
1905: @item return
1906: Number
1907: @item p
1908: Number
1909: @item s
1910: Number
1911: @end table
1912:
1913: @itemize @bullet
1914: @item If @var{s} is 1, then all Grobner bases to be computed
1915: will be the reduced Grobner bases.
1916: @item If @var{s} is 0, then Grobner bases are not necessarily the reduced
1917: Grobner bases. This is the default.
1918: @end itemize
1919: */
1920:
1921: /*&jp-texi
1.8 takayama 1922: @node sm1.auto_reduce,,, SM1 Functions
1923: @subsection @code{sm1.auto_reduce}
1924: @findex sm1.auto_reduce
1.1 takayama 1925: @table @t
1.8 takayama 1926: @item sm1.auto_reduce(@var{s}|proc=@var{p})
1.1 takayama 1927: :: $B%U%i%0(B "AutoReduce" $B$r(B @var{s} $B$K@_Dj(B.
1928: @end table
1929:
1930: @table @var
1931: @item $BLa$jCM(B
1932: $B?t(B
1933: @item p
1934: $B?t(B
1935: @item s
1936: $B?t(B
1937: @end table
1938:
1939: @itemize @bullet
1940: @item @var{s} $B$,(B 1 $B$N$H$-(B, $B0J8e7W;;$5$l$k%0%l%V%J4pDl$O$9$Y$F(B,
1941: reduced $B%0%l%V%J4pDl$H$J$k(B.
1942: @item @var{s} $B$,(B 0 $B$N$H$-(B, $B7W;;$5$l$k%0%l%V%J4pDl$O(B
1943: reduced $B%0%l%V%J4pDl$H$O$+$.$i$J$$(B. $B$3$A$i$,%G%U%)!<%k%H(B.
1944: @end itemize
1945: */
1946:
1947:
1948:
1949: /*&eg-texi
1.8 takayama 1950: @node sm1.slope,,, SM1 Functions
1951: @subsection @code{sm1.slope}
1952: @findex sm1.slope
1.1 takayama 1953: @table @t
1.8 takayama 1954: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
1.1 takayama 1955: :: Returns the slopes of differential equations @var{ii}.
1956: @end table
1957:
1958: @table @var
1959: @item return
1960: List
1961: @item p
1962: Number
1963: @item ii
1964: List (equations)
1965: @item v
1966: List (variables)
1967: @item f_filtration
1968: List (weight vector)
1969: @item v_filtration
1970: List (weight vector)
1971: @end table
1972:
1973: @itemize @bullet
1.8 takayama 1974: @item @code{sm1.slope} returns the (geometric) slopes
1.1 takayama 1975: of the system of differential equations @var{ii}
1976: along the hyperplane specified by
1977: the V filtration @var{v_filtration}.
1978: @item @var{v} is a list of variables.
1979: @item The return value is a list of lists.
1980: The first entry of each list is the slope and the second entry
1981: is the weight vector for which the microcharacteristic variety is
1982: not bihomogeneous.
1983: @end itemize
1.5 takayama 1984:
1985: @noindent
1986: Algorithm:
1987: see "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
1988: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
1989: Note that the signs of the slopes are negative, but the absolute values
1990: of the slopes are returned.
1991:
1.1 takayama 1992: */
1993:
1994: /*&jp-texi
1.8 takayama 1995: @node sm1.slope,,, SM1 Functions
1996: @subsection @code{sm1.slope}
1997: @findex sm1.slope
1.1 takayama 1998: @table @t
1.8 takayama 1999: @item sm1.slope(@var{ii},@var{v},@var{f_filtration},@var{v_filtration}|proc=@var{p})
1.1 takayama 2000: :: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B slope $B$rLa$9(B.
2001: @end table
2002:
2003: @table @var
2004: @item return
2005: $B?t(B
2006: @item p
2007: $B?t(B
2008: @item ii
2009: $B%j%9%H(B ($BJ}Dx<0(B)
2010: @item v
2011: $B%j%9%H(B ($BJQ?t(B)
2012: @item f_filtration
2013: $B%j%9%H(B (weight vector)
2014: @item v_filtration
2015: $B%j%9%H(B (weight vector)
2016: @end table
2017:
2018: @itemize @bullet
1.8 takayama 2019: @item @code{sm1.slope} $B$O(B
1.1 takayama 2020: $BHyJ,J}Dx<07O(B @var{ii} $B$N(B V filtration @var{v_filtration}
2021: $B$G;XDj$9$kD6J?LL$K1h$C$F$N(B (geomeric) slope $B$r7W;;$9$k(B.
2022: @item @var{v} $B$OJQ?t$N%j%9%H(B.
1.5 takayama 2023: @item $BLa$jCM$O(B, $B%j%9%H$r@.J,$H$9$k%j%9%H$G$"$k(B.
2024: $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
2025: microcharacteristic variety $B$,(B bihomogeneous $B$G$J$$(B.
2026: @end itemize
2027:
2028: @noindent
2029: Algorithm:
1.1 takayama 2030: "A.Assi, F.J.Castro-Jimenez and J.M.Granger,
2031: How to calculate the slopes of a D-module, Compositio Math, 104, 1-17, 1996"
2032: $B$r$_$h(B.
2033: Slope $B$NK\Mh$NDj5A$G$O(B, $BId9f$,Ii$H$J$k$,(B, $B$3$N%W%m%0%i%`$O(B,
2034: Slope $B$N@dBPCM$rLa$9(B.
2035: */
2036:
2037: /*&C-texi
2038:
2039: @example
2040:
1.8 takayama 2041: [284] A= sm1.gkz([ [[1,2,3]], [-3] ]);
1.1 takayama 2042:
2043:
1.8 takayama 2044: [285] sm1.slope(A[0],A[1],[0,0,0,1,1,1],[0,0,-1,0,0,1]);
1.1 takayama 2045:
1.8 takayama 2046: [286] A2 = sm1.gkz([ [[1,1,1,0],[2,-3,1,-3]], [1,0]]);
1.1 takayama 2047: (* This is an interesting example given by Laura Matusevich,
2048: June 9, 2001 *)
2049:
1.8 takayama 2050: [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 2051:
2052:
2053: @end example
2054:
2055: */
2056: /*&eg-texi
2057: @table @t
2058: @item Reference
1.10 ! takayama 2059: @code{sm.gb}
1.1 takayama 2060: @end table
2061: */
2062: /*&jp-texi
2063: @table @t
2064: @item $B;2>H(B
1.10 ! takayama 2065: @code{sm.gb}
1.1 takayama 2066: @end table
1.4 takayama 2067: */
2068:
2069:
2070: /*&eg-texi
2071: @include sm1-auto-en.texi
2072: */
2073:
2074: /*&jp-texi
2075: @include sm1-auto-ja.texi
1.1 takayama 2076: */
2077:
2078:
2079: end$
2080:
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