Annotation of OpenXM/src/kan96xx/Doc/hol.sm1, Revision 1.8
1.8 ! takayama 1: % $OpenXM: OpenXM/src/kan96xx/Doc/hol.sm1,v 1.7 2000/06/12 08:03:56 takayama Exp $
1.5 takayama 2: %% hol.sm1, 1998, 11/8, 11/10, 11/14, 11/25, 1999, 5/18, 6/5. 2000, 6/8
1.1 maekawa 3: %% rank, rrank, characteristic
4: %% This file is error clean.
5: /hol.version (2.990515) def
6: hol.version [(Version)] system_variable gt
7: { [(This package hol.sm1 requires the latest version of kan/sm1) nl
8: (Please get it from http://www.math.kobe-u.ac.jp/KAN)
9: ] cat
10: error
11: } { } ifelse
12:
1.5 takayama 13: $hol.sm1, basic package for holonomic systems (C) N.Takayama, 2000, 06/08 $
1.1 maekawa 14: message-quiet
15:
1.7 takayama 16: /gb.warning 0 def
1.1 maekawa 17: /rank.v [(x) (y) (z)] def %% default value of v (variables).
18: /rank.ch [ ] def %% characteristic variety.
19: /rank.verbose 0 def
20: /rank {
21: /arg1 set
22: [/in-rank /aa /typev /setarg /f /v /vsss /vddd
23: /gg /wv /vd /vdweight /chv
24: /one
25: ] pushVariables
26: [(CurrentRingp) (KanGBmessage)] pushEnv
27: [
28:
29: /aa arg1 def
30: aa isArray { } { ( << array >> rank) error } ifelse
31: /setarg 0 def
32: aa { tag } map /typev set
33: typev [ ArrayP ] eq
34: { /f aa 0 get def
35: /v rank.v def
36: /setarg 1 def
37: } { } ifelse
38: typev [ArrayP StringP] eq
39: { /f aa 0 get def
40: /v [ aa 1 get to_records pop ] def
41: /setarg 1 def
42: } { } ifelse
43: typev [ArrayP ArrayP] eq
44: { /f aa 0 get def
45: /v aa 1 get def
46: /setarg 1 def
47: } { } ifelse
48: setarg { } { (rank : Argument mismatch) error } ifelse
49:
50: [(KanGBmessage) rank.verbose ] system_variable
51:
52: f { toString } map /f set
53: v { @@@.Dsymbol 2 1 roll 2 cat_n 1 } map
54: /vddd set %% vddd = [(Dx) 1 (Dy) 1 (Dz) 1]
55: v { @@@.Dsymbol 2 1 roll 2 cat_n } map
56: /vd set %% vd = [(Dx) (Dy) (Dz)]
57: /vdweight
58: vd { [ 2 1 roll -1 ] } map %% vdweight=[[(Dx) -1] [(Dy) -1] [(Dz) -1]]
59: def
60:
61: [v from_records
62: ring_of_differential_operators [vddd] weight_vector 0] define_ring
63: f { . dehomogenize } map /f set
64: [f] groebner_sugar 0 get /gg set
65:
66: /wv vddd weightv def
67: gg { wv init } map /chv set %%obtained the characteristic variety.
68: /rank.ch chv def
69: chv { toString } map /chv set
70:
71: [ v vd join from_records
72: ring_of_polynomials
73: [vddd] vdweight join weight_vector
74: 0
75: ] define_ring
76: [chv {.} map] groebner_sugar 0 get { init } map /chii set
77:
78: /rank.chii chii def
79: rank.verbose { chii message } { } ifelse
80: v {[ 2 1 roll . (1).]} map /one set
81: %% [[(x). (1).] [(y). (1).] [(z). (1).]]
82: %% chii { one replace } map %% buggy code.
83: %% Arg of hilb should be a reduced GB.
84: [chii { one replace } map] groebner 0 get
85: vd hilb /arg1 set
86: ] pop
87: popEnv
88: popVariables
89: arg1
90: } def
91:
92:
93: [(rank)
94: [( a rank b)
95: ( array a; number b)
96: (Example 1 : )
97: $ [ [( (x Dx)^2 + ( y Dy)^2) ( x Dx y Dy -1)] (x,y)] rank :: $
98: (Example 2 : )
99: $[ [( (x^3-y^2) Dx + 3 x^2) ( (x^3-y^2) Dy - 2 y)] (x,y)] rank :: $
100: ]
101: ] putUsages
102: (rank ) messagen-quiet
103:
104: /characteristic.verbose 0 def
105: /characteristic.v [(x) (y) (z)] def
106: /characteristic.ch [ ] def
107: /ch { characteristic } def
108: /characteristic {
109: /arg1 set
110: [/in-rank /aa /typev /setarg /f /v /vsss /vddd
111: /gg /wv /vd /chv
112: /one
113: ] pushVariables
114: [(CurrentRingp) (KanGBmessage)] pushEnv
115: [
116:
117: /aa arg1 def
118: aa isArray { } { ( << array >> characteristic) error } ifelse
119: /setarg 0 def
120: aa { tag } map /typev set
121: typev [ ArrayP ] eq
122: { /f aa 0 get def
123: /v characteristic.v def
124: /setarg 1 def
125: } { } ifelse
126: typev [ArrayP StringP] eq
127: { /f aa 0 get def
128: /v [ aa 1 get to_records pop ] def
129: /setarg 1 def
130: } { } ifelse
131: typev [ArrayP ArrayP] eq
132: { /f aa 0 get def
133: /v aa 1 get def
134: /setarg 1 def
135: } { } ifelse
136: setarg { } { (rank : Argument mismatch) error } ifelse
137:
138: [(KanGBmessage) characteristic.verbose ] system_variable
139:
140: f { toString } map /f set
141: v { @@@.Dsymbol 2 1 roll 2 cat_n 1 } map
142: /vddd set %% vddd = [(Dx) 1 (Dy) 1 (Dz) 1]
143: v { @@@.Dsymbol 2 1 roll 2 cat_n } map
144: /vd set %% vd = [(Dx) (Dy) (Dz)]
145:
146: [v from_records
147: ring_of_differential_operators [vddd] weight_vector 0] define_ring
148: f { . dehomogenize } map /f set
149: [f] groebner_sugar 0 get /gg set
150:
151: /wv vddd weightv def
152: gg { wv init } map /chv set
153: /characteristic.ch [chv] def
154: %% gg { wv init toString} map /chv set %%obtained the characteristic variety.
155: %% /characteristic.ch chv def
156:
157: %% [ v vd join from_records
158: %% ring_of_polynomials
159: %% [vddd] weight_vector
160: %% 0
161: %% ] define_ring
162: %% [chv {.} map] groebner_sugar 0 get /characteristic.ch set
163:
164: characteristic.ch /arg1 set
165: ] pop
166: popEnv
167: popVariables
168: arg1
169: } def
170:
171: [(characteristic)
172: [( a characteristic b)
173: ( array a; number b)
174: (b is the generator of the characteristic variety of a.)
175: (For the algorithm, see Japan J. of Industrial and Applied Math., 1994, 485--497.)
176: (Example 1 : )
177: $ [ [( (x Dx)^2 + ( y Dy)^2) ( x Dx y Dy -1)] (x,y)] characteristic :: $
178: (Example 2 : )
179: $[ [( (x^3-y^2) Dx + 3 x^2) ( (x^3-y^2) Dy - 2 y)] (x,y)] characteristic :: $
180: ]
181: ] putUsages
182: (characteristic ) messagen-quiet
183: [(ch)
184: [(ch is the abbreviation of characteristic.)
185: ( a ch b)
186: ( array a; number b)
187: (b is the generator of the characteristic variety of a.)
188: (For the algorithm, see, Japan J. of Industrial and Applied Math., 1994, 485--497.)
189: (Example 1 : )
190: $ [ [( (x Dx)^2 + ( y Dy)^2) ( x Dx y Dy -1)] (x,y)] ch :: $
191: (Example 2 : )
192: $[ [( (x^3-y^2) Dx + 3 x^2) ( (x^3-y^2) Dy - 2 y)] (x,y)] ch :: $
193: ]
194: ] putUsages
195: (ch ) messagen-quiet
196:
197: %%%% developing rrank.sm1
198: /rrank.v [(x) (y) (z)] def %% default value of v (variables).
199: /rrank.init [ ] def %% initial ideal.
200: /rrank.verbose 0 def
201: /rrank {
202: /arg1 set
203: [/in-rrank /aa /typev /setarg /f /v /vsss /vddd
204: /gg /wv /vd /vdweight
205: /one /i /chv
206: ] pushVariables
207: [(CurrentRingp) (KanGBmessage)] pushEnv
208: [
209:
210: /aa arg1 def
211: aa isArray { } { ( << array >> rrank) error } ifelse
212: /setarg 0 def
213: aa { tag } map /typev set
214: typev [ ArrayP ] eq
215: { /f aa 0 get def
216: /v rrank.v def
217: /setarg 1 def
218: } { } ifelse
219: typev [ArrayP StringP] eq
220: { /f aa 0 get def
221: /v [ aa 1 get to_records pop ] def
222: /setarg 1 def
223: } { } ifelse
224: typev [ArrayP ArrayP] eq
225: { /f aa 0 get def
226: /v aa 1 get def
227: /setarg 1 def
228: } { } ifelse
229: setarg { } { (rrank : Argument mismatch) error } ifelse
230:
231: [(KanGBmessage) rrank.verbose ] system_variable
232:
233: f { toString } map /f set
234: v { @@@.Dsymbol 2 1 roll 2 cat_n 1 } map
235:
236: v { @@@.Dsymbol 2 1 roll 2 cat_n } map
237: /vd set %% vd = [(Dx) (Dy) (Dz)] , v = [(x) (y) (z)]
238: /vdweight
239: [ 0 1 v length 1 sub { /i set v i get << 0 i sub >>
240: vd i get << i >> } for ]
241: def
242: rrank.verbose { vdweight message } { } ifelse
243:
244: [v from_records
245: ring_of_differential_operators [vdweight] weight_vector 0] define_ring
246: f { . dehomogenize homogenize } map /f set
247: [f] groebner 0 get {dehomogenize} map /gg set
248:
249: /wv vdweight weightv def
250: gg { wv init } map /rrank.init set %%obtained the initial ideal
251: rrank.init {toString} map /chv set
252: /arg1 [chv v] rank def
253: ] pop
254: popEnv
255: popVariables
256: arg1
257: } def
258:
259:
260: [(rrank)
261: [( a rrank b)
262: ( array a; number b)
263: (It computes the holonomic rank for regular holonomic system.)
264: (For the algorithm, see Grobner deformations of hypergeometric differential equations, 1999, Springer.)
265: (Chapter 2.)
266: (Example 1 : )
267: $ [ [( (x Dx)^2 + ( y Dy)^2) ( x Dx y Dy -1)] (x,y)] rrank :: $
268: ]
269: ] putUsages
270: (rrank ) messagen-quiet
271:
272: /gb.v 1 def
273: /gb.verbose 0 def
1.4 takayama 274: /gb.options [ ] def
1.1 maekawa 275: /gb {
276: /arg1 set
277: [/in-gb /aa /typev /setarg /f /v
278: /gg /wv /termorder /vec /ans /rr /mm
279: ] pushVariables
280: [(CurrentRingp) (KanGBmessage)] pushEnv
281: [
282:
283: /aa arg1 def
284: aa isArray { } { ( << array >> gb) error } ifelse
285: /setarg 0 def
286: /wv 0 def
287: aa { tag } map /typev set
288: typev [ ArrayP ] eq
289: { /f aa 0 get def
290: /v gb.v def
291: /setarg 1 def
292: } { } ifelse
293: typev [ArrayP StringP] eq
294: { /f aa 0 get def
295: /v aa 1 get def
296: /setarg 1 def
297: } { } ifelse
298: typev [ArrayP ArrayP] eq
299: { /f aa 0 get def
300: /v aa 1 get from_records def
301: /setarg 1 def
302: } { } ifelse
303: typev [ArrayP StringP ArrayP] eq
304: { /f aa 0 get def
305: /v aa 1 get def
306: /wv aa 2 get def
307: /setarg 1 def
308: } { } ifelse
309: typev [ArrayP ArrayP ArrayP] eq
310: { /f aa 0 get def
311: /v aa 1 get from_records def
312: /wv aa 2 get def
313: /setarg 1 def
314: } { } ifelse
315:
316: setarg { } { (gb : Argument mismatch) error } ifelse
317:
318: [(KanGBmessage) gb.verbose ] system_variable
319:
320: %%% Start of the preprocess
321: f getRing /rr set
322: %% To the normal form : matrix expression.
323: f gb.toMatrixOfString /f set
324: /mm gb.itWasMatrix def
325:
326: rr tag 0 eq {
327: %% Define our own ring
328: v isInteger {
329: (Error in gb: Specify variables) error
330: } { } ifelse
331: wv isInteger {
332: [v ring_of_differential_operators
333: 0] define_ring
334: /termorder 1 def
335: }{
336: [v ring_of_differential_operators
337: wv weight_vector
338: 0] define_ring
339: wv gb.isTermOrder /termorder set
340: } ifelse
341: } {
342: %% Use the ring structre given by the input.
343: v isInteger not {
1.7 takayama 344: gb.warning {
345: (Warning : the given ring definition is not used.) message
346: } { } ifelse
1.1 maekawa 347: } { } ifelse
348: rr ring_def
349: /wv rr gb.getWeight def
350: wv gb.isTermOrder /termorder set
351: } ifelse
352: %%% Enf of the preprocess
353:
1.4 takayama 354: gb.verbose { (gb.options = ) messagen gb.options message } { } ifelse
1.1 maekawa 355: termorder {
356: f { {. dehomogenize} map } map /f set
1.4 takayama 357: [f gb.options] groebner_sugar 0 get /gg set
1.1 maekawa 358: }{
359: f { {. dehomogenize} map} map /f set
360: f fromVectors { homogenize } map /f set
1.4 takayama 361: [f gb.options] groebner 0 get /gg set
1.1 maekawa 362: }ifelse
363: wv isInteger {
364: /ans [gg gg {init} map] def
365: }{
366: /ans [gg gg {wv 0 get weightv init} map] def
367: }ifelse
368:
369: %% Postprocess : recover the matrix expression.
370: mm {
371: ans { /tmp set [mm tmp] toVectors } map
372: /ans set
373: }{ }
374: ifelse
375: %%
376:
377: /arg1 ans def
378: ] pop
379: popEnv
380: popVariables
381: arg1
382: } def
383: (gb ) messagen-quiet
384:
385: /pgb {
386: /arg1 set
387: [/in-pgb /aa /typev /setarg /f /v
388: /gg /wv /termorder /vec /ans /rr /mm
389: ] pushVariables
390: [(CurrentRingp) (KanGBmessage) (UseCriterion1)] pushEnv
391: [
392:
393: /aa arg1 def
394: aa isArray { } { (<< array >> pgb) error } ifelse
395: /setarg 0 def
396: /wv 0 def
397: aa { tag } map /typev set
398: typev [ ArrayP ] eq
399: { /f aa 0 get def
400: /v gb.v def
401: /setarg 1 def
402: } { } ifelse
403: typev [ArrayP StringP] eq
404: { /f aa 0 get def
405: /v aa 1 get def
406: /setarg 1 def
407: } { } ifelse
408: typev [ArrayP ArrayP] eq
409: { /f aa 0 get def
410: /v aa 1 get from_records def
411: /setarg 1 def
412: } { } ifelse
413: typev [ArrayP StringP ArrayP] eq
414: { /f aa 0 get def
415: /v aa 1 get def
416: /wv aa 2 get def
417: /setarg 1 def
418: } { } ifelse
419: typev [ArrayP ArrayP ArrayP] eq
420: { /f aa 0 get def
421: /v aa 1 get from_records def
422: /wv aa 2 get def
423: /setarg 1 def
424: } { } ifelse
425:
426: setarg { } { (pgb : Argument mismatch) error } ifelse
427:
428: [(KanGBmessage) gb.verbose ] system_variable
429:
430: %%% Start of the preprocess
431: f getRing /rr set
432: %% To the normal form : matrix expression.
433: f gb.toMatrixOfString /f set
434: /mm gb.itWasMatrix def
435:
436: rr tag 0 eq {
437: %% Define our own ring
438: v isInteger {
439: (Error in pgb: Specify variables) error
440: } { } ifelse
441: wv isInteger {
442: [v ring_of_polynomials
443: 0] define_ring
444: /termorder 1 def
445: }{
446: [v ring_of_polynomials
447: wv weight_vector
448: 0] define_ring
449: wv gb.isTermOrder /termorder set
450: } ifelse
451: } {
452: %% Use the ring structre given by the input.
453: v isInteger not {
1.7 takayama 454: gb.warning {
455: (Warning : the given ring definition is not used.) message
456: } { } ifelse
1.1 maekawa 457: } { } ifelse
458: rr ring_def
459: /wv rr gb.getWeight def
460: wv gb.isTermOrder /termorder set
461: } ifelse
462: %%% Enf of the preprocess
463:
1.4 takayama 464: gb.verbose { (gb.options = ) messagen gb.options message } { } ifelse
1.1 maekawa 465: termorder {
466: f { {. dehomogenize} map } map /f set
467: [(UseCriterion1) 1] system_variable
1.4 takayama 468: [f gb.options] groebner_sugar 0 get /gg set
1.1 maekawa 469: [(UseCriterion1) 0] system_variable
470: }{
471: f { {. dehomogenize} map} map /f set
472: f fromVectors { homogenize } map /f set
473: [(UseCriterion1) 1] system_variable
1.4 takayama 474: [f gb.options] groebner 0 get /gg set
1.1 maekawa 475: [(UseCriterion1) 0] system_variable
476: }ifelse
477: wv isInteger {
478: /ans [gg gg {init} map] def
479: }{
480: /ans [gg gg {wv 0 get weightv init} map] def
481: }ifelse
482:
483: %% Postprocess : recover the matrix expression.
484: mm {
485: ans { /tmp set [mm tmp] toVectors } map
486: /ans set
487: }{ }
488: ifelse
489: %%
490:
491: /arg1 ans def
492: ] pop
493: popEnv
494: popVariables
495: arg1
496: } def
497:
498: /pgb.old {
499: /arg1 set
500: [/in-pgb /aa /typev /setarg /f /v
501: /gg /wv /termorder /vec /ans
502: ] pushVariables
503: [(CurrentRingp) (KanGBmessage) (UseCriterion1)] pushEnv
504: [
505:
506: /aa arg1 def
507: aa isArray { } { (array pgb) message (pgb) usage error } ifelse
508: /setarg 0 def
509: /wv 0 def
510: aa { tag } map /typev set
511: typev [ ArrayP ] eq
512: { /f aa 0 get def
513: /v gb.v def
514: /setarg 1 def
515: } { } ifelse
516: typev [ArrayP StringP] eq
517: { /f aa 0 get def
518: /v aa 1 get def
519: /setarg 1 def
520: } { } ifelse
521: typev [ArrayP ArrayP] eq
522: { /f aa 0 get def
523: /v aa 1 get from_records def
524: /setarg 1 def
525: } { } ifelse
526: typev [ArrayP StringP ArrayP] eq
527: { /f aa 0 get def
528: /v aa 1 get def
529: /wv aa 2 get def
530: /setarg 1 def
531: } { } ifelse
532: typev [ArrayP ArrayP ArrayP] eq
533: { /f aa 0 get def
534: /v aa 1 get from_records def
535: /wv aa 2 get def
536: /setarg 1 def
537: } { } ifelse
538:
539: setarg { } { (pgb : Argument mismatch) message error } ifelse
540:
541: [(KanGBmessage) gb.verbose ] system_variable
542:
543: %% Input must not be vectors.
544: f { toString } map /f set
545:
546: wv isInteger {
547: [v ring_of_polynomials
548: 0] define_ring
549: /termorder 1 def
550: }{
551: [v ring_of_polynomials
552: wv weight_vector
553: 0] define_ring
554: wv gb.isTermOrder /termorder set
555: } ifelse
556: termorder {
557: f { . dehomogenize } map /f set
558: [(UseCriterion1) 1] system_variable
559: [f] groebner_sugar 0 get /gg set
560: [(UseCriterion1) 0] system_variable
561: }{
562: f { . dehomogenize homogenize} map /f set
563: [(UseCriterion1) 1] system_variable
564: [f] groebner 0 get /gg set
565: [(UseCriterion1) 0] system_variable
566: }ifelse
567: wv isInteger {
568: /ans [gg gg {init} map] def
569: }{
570: /ans [gg gg {wv 0 get weightv init} map] def
571: }ifelse
572: /arg1 ans def
573: ] pop
574: popEnv
575: popVariables
576: arg1
577: } def
578: (pgb ) messagen-quiet
579:
580: /gb.toMatrixOfString {
581: /arg1 set
582: [/in-gb.toMatrixOfString /ff /aa /ans] pushVariables
583: [
584: /aa arg1 def
585: aa length 0 eq { /ans [ ] def /gb.toMatrixOfString.LLL goto }{ } ifelse
586: aa 0 get isArray {
587: /gb.itWasMatrix aa 0 get length def
588: }{
589: /gb.itWasMatrix 0 def
590: } ifelse
591: aa {
592: /ff set
593: ff isArray {
594: ff {toString} map /ff set
595: }{
596: [ff toString] /ff set
597: } ifelse
598: ff
599: } map /ans set
600: /gb.toMatrixOfString.LLL
601: /arg1 ans def
602: ] pop
603: popVariables
604: arg1
605: } def
606: [(gb.toMatrixOfString)
607: [(It translates given input into a matrix form which is a data structure)
608: (for computations of kernel, image, cokernel, etc.)
609: (gb.itWasMatrix is set to the length of the input vector.)
610: $Example 1: $
611: $ [ (x). (y).] gb.toMatrixOfString ==> [[(x)] [(y)]] $
612: $ gb.itWasMatrix is 0.$
613: $Example 2: $
614: $ [ [(x). (1).] [(y). (0).]] gb.toMatrixOfString ==> [ [(x) (1)] [(y) (0)]] $
615: $ gb.itWasMatrix is 2.$
616: ]] putUsages
617:
618: /gb.toMatrixOfPoly {
619: /arg1 set
620: [/in-gb.toMatrixOfPoly /ff /aa /ans] pushVariables
621: [
622: /aa arg1 def
623: aa length 0 eq { /ans [ ] def /gb.toMatrixOfPoly.LLL goto }{ } ifelse
624: aa 0 get isArray {
625: /gb.itWasMatrix aa 0 get length def
626: }{
627: /gb.itWasMatrix 0 def
628: } ifelse
629: aa {
630: /ff set
631: ff isArray {
632: }{
633: [ff] /ff set
634: } ifelse
635: ff
636: } map /ans set
637: /gb.toMatrixOfPoly.LLL
638: /arg1 ans def
639: ] pop
640: popVariables
641: arg1
642: } def
643: [(gb.toMatrixOfPoly)
644: [(It translates given input into a matrix form which is a data structure)
645: (for computations of kernel, image, cokernel, etc.)
646: (gb.itWasMatrix is set to the length of the input vector.)
647: $Example 1: $
648: $ [ (x). (y).] gb.toMatrixOfPoly ==> [[(x)] [(y)]] $
649: $ gb.itWasMatrix is 0.$
650: $Example 2: $
651: $ [ [(x). (1).] [(y). (0).]] gb.toMatrixOfPoly ==> [ [(x) (1)] [(y) (0)]] $
652: $ gb.itWasMatrix is 2.$
653: ]] putUsages
654:
655: /gb.getWeight {
656: /arg1 set
657: [/in-gb.getWeight /rr /ww /vv /ans /nn /ii] pushVariables
658: [(CurrentRingp)] pushEnv
659: [
660: /rr arg1 def
661: rr ring_def
662: getVariableNames /vv set
663: [(orderMatrix)] system_variable 0 get /ww set
664: /nn vv length 1 sub def
665: [0 1 nn {
666: /ii set
667: ww ii get 0 eq {
668: } {
669: vv ii get
670: ww ii get
671: } ifelse
672: } for
673: ] /ans set
674: /arg1 [ans] def
675: ] pop
676: popEnv
677: popVariables
678: arg1
679: } def
680: [(gb.getWeight)
681: [(ring gb.getWeight wv)
682: (It gets the weight vector field of the ring ring.)
683: ]] putUsages
684:
685:
686: /gb.isTermOrder {
687: /arg1 set
688: [/in-gb.isTermOrder /vv /ww /yes /i /j] pushVariables
689: [
690: /vv arg1 def
691: /yes 1 def
692: 0 1 vv length 1 sub {
693: /i set
694: /ww vv i get def
695: 0 1 ww length 1 sub {
696: /j set
697: ww j get isInteger {
698: ww j get 0 lt { /yes 0 def } { } ifelse
699: }{ } ifelse
700: }for
701: }for
702: /arg1 yes def
703: ] pop
704: popVariables
705: arg1
706: } def
707: [(gb)
708: [(a gb b)
709: (array a; array b;)
710: (b : [g ii]; array g; array in; g is a Grobner basis of f)
711: ( in the ring of differential operators.)
712: $ ii is the initial ideal in case of w is given or <<a>> belongs$
713: $ to a ring. In the other cases, it returns the initial monominal.$
714: (a : [f ]; array f; f is a set of generators of an ideal in a ring.)
715: (a : [f v]; array f; string v; v is the variables. )
716: (a : [f v w]; array f; string v; array of array w; w is the weight matirx.)
717: ( )
718: $Example 1: [ [( (x Dx)^2 + (y Dy)^2 -1) ( x y Dx Dy -1)] (x,y) $
719: $ [ [ (Dx) 1 ] ] ] gb pmat ; $
720: (Example 2: )
721: (To put h=1, type in, e.g., )
722: $ [ [(2 x Dx + 3 y Dy+6) (2 y Dx + 3 x^2 Dy)] (x,y) $
723: $ [[(x) -1 (Dx) 1 (y) -1 (Dy) 1]]] gb /gg set gg dehomogenize pmat ;$
724: ( )
725: $Example 3: [ [( (x Dx)^2 + (y Dy)^2 -1) ( x y Dx Dy -1)] (x,y) $
726: $ [ [ (Dx) 1 (Dy) 1] ] ] gb pmat ; $
727: ( )
728: $Example 4: [[ [(x^2) (y+x)] [(x+y) (y^3)] [(2 x^2+x y) (y+x+x y^3)]] (x,y) $
729: $ [ [ (x) -1 (y) -1] ] ] gb pmat ; $
730: ( )
731: (cf. gb, groebner, groebner_sugar, syz. )
732: ]] putUsages
733:
734: [(pgb)
735: [(a pgb b)
736: (array a; array b;)
737: (b : [g ii]; array g; array in; g is a Grobner basis of f)
738: ( in the ring of polynomials.)
739: $ ii is the initial ideal in case of w is given or <<a>>belongs$
740: $ to a ring. In the other cases, it returns the initial monominal.$
741: (a : [f ]; array f; f is a set of generators of an ideal in a ring.)
742: (a : [f v]; array f; string v; v is the variables.)
743: (a : [f v w]; array f; string v; array of array w; w is the weight matirx.)
744: $Example 1: [(x,y) ring_of_polynomials 0] define_ring $
745: $ [ [(x^2+y^2-4). (x y -1).] ] pgb :: $
746: $Example 2: [ [(x^2+y^2) (x y)] (x,y) [ [(x) -1 (y) -1] ] ] pgb :: $
747: (cf. gb, groebner, groebner_sugar, syz. )
748: ]] putUsages
749:
750:
751: %/syz.v 1 def
752: /syz.v 1 def
753: /syz.verbose 0 def
754: /syz {
755: /arg1 set
756: [/in-syz /aa /typev /setarg /f /v
757: /gg /wv /termorder /vec /ans /ggall /vectorInput /vsize /gtmp /gtmp2
758: /rr /mm
759: ] pushVariables
760: [(CurrentRingp) (KanGBmessage)] pushEnv
761: [
762:
763: /aa arg1 def
764: aa isArray { } { (<< array >> syz) error } ifelse
765: /setarg 0 def
766: /wv 0 def
767: aa { tag } map /typev set
768: typev [ ArrayP ] eq
769: { /f aa 0 get def
770: /v syz.v def
771: /setarg 1 def
772: } { } ifelse
773: typev [ArrayP StringP] eq
774: { /f aa 0 get def
775: /v aa 1 get def
776: /setarg 1 def
777: } { } ifelse
778: typev [ArrayP ArrayP] eq
779: { /f aa 0 get def
780: /v aa 1 get from_records def
781: /setarg 1 def
782: } { } ifelse
783: typev [ArrayP StringP ArrayP] eq
784: { /f aa 0 get def
785: /v aa 1 get def
786: /wv aa 2 get def
787: /setarg 1 def
788: } { } ifelse
789: typev [ArrayP ArrayP ArrayP] eq
790: { /f aa 0 get def
791: /v aa 1 get from_records def
792: /wv aa 2 get def
793: /setarg 1 def
794: } { } ifelse
795:
796: setarg { } { (syz : Argument mismatch) error } ifelse
797:
798: [(KanGBmessage) syz.verbose ] system_variable
799:
800:
801:
802: %%% Start of the preprocess
803: f getRing /rr set
804: %% To the normal form : matrix expression.
805: f gb.toMatrixOfString /f set
806: /mm gb.itWasMatrix def
807: mm 0 gt {
808: /vectorInput 1 def
809: }{
810: /vectorInput 1 def
811: } ifelse
812:
813: rr tag 0 eq {
814: %% Define our own ring
815: v isInteger {
816: (Error in syz: Specify variables) error
817: } { } ifelse
818: wv isInteger {
819: [v ring_of_differential_operators
820: 0] define_ring
821: /termorder 1 def
822: }{
823: [v ring_of_differential_operators
824: wv weight_vector
825: 0] define_ring
826: wv gb.isTermOrder /termorder set
827: } ifelse
828: }{
829: %% Use the ring structre given by the input.
830: v isInteger not {
1.7 takayama 831: gb.warning {
832: (Warning : the given ring definition is not used.) message
833: } { } ifelse
1.1 maekawa 834: } { } ifelse
835: rr ring_def
836: /wv rr gb.getWeight def
837: wv gb.isTermOrder /termorder set
838: } ifelse
839: %%% Enf of the preprocess
840:
841: termorder {
842: f { {. dehomogenize} map } map /f set
843: [f [(needBack) (needSyz)]] groebner_sugar /ggall set
844: ggall 2 get /gg set
845: }{
846: f { {. dehomogenize } map homogenize } map /f set
847: [f [(needBack) (needSyz)]] groebner /ggall set
848: ggall 2 get /gg set
849: }ifelse
850: vectorInput {
851: /vsize f 0 get length def %% input vector size.
852: /gtmp ggall 0 get def
853: [vsize gtmp] toVectors /gtmp set
854: ggall 0 gtmp put
855: }{ } ifelse
856: /arg1 [gg dehomogenize ggall] def
857: ] pop
858: popEnv
859: popVariables
860: arg1
861: } def
862: (syz ) messagen-quiet
863:
864: [(syz)
865: [(a syz [b c])
866: (array a; array b; array c)
867: (b is a set of generators of the syzygies of f.)
868: (c = [gb, backward transformation, syzygy without dehomogenization].)
869: (See groebner.)
870: (a : [f ]; array f; f is a set of generators of an ideal in a ring.)
871: (a : [f v]; array f; string v; v is the variables.)
872: (a : [f v w]; array f; string v; array of array w; w is the weight matirx.)
873: $Example 1: [(x,y) ring_of_polynomials 0] define_ring $
874: $ [ [(x^2+y^2-4). (x y -1).] ] syz :: $
875: $Example 2: [ [(x^2+y^2) (x y)] (x,y) [ [(x) -1 (y) -1] ] ] syz :: $
876: $Example 3: [ [( (x Dx)^2 + (y Dy)^2 -1) ( x y Dx Dy -1)] (x,y) $
877: $ [ [ (Dx) 1 ] ] ] syz pmat ; $
878: $Example 4: [ [(2 x Dx + 3 y Dy+6) (2 y Dx + 3 x^2 Dy)] (x,y) $
879: $ [[(x) -1 (Dx) 1 (y) -1 (Dy) 1]]] syz pmat ;$
880: $Example 5: [ [ [(x^2) (y+x)] [(x+y) (y^3)] [(2 x^2+x y) (y+x+x y^3)]] $
881: $ (x,y) ] syz pmat ;$
882: $Example 6: [ [ [(x^2) (y+x)] [(x+y) (y^3)] [(2 x^2+x y) (y+x+x y^3)]] $
883: $ (x,y) [[(x) -1 (y) -2]] ] syz pmat ;$
884: $Example 7: [ [ [(0) (0)] [(0) (0)] [(x) (y)]] $
885: $ [(x) (y)]] syz pmat ;$
886: ]] putUsages
887:
888:
889: %%%%%%%%%%%%%%%%%% package fs %%%%%%%%%%%%%%%%%%%%%%%
890: [(genericAnn)
891: [ (f [s v1 v2 ... vn] genericAnn [L1 ... Lm])
892: (L1, ..., Lm are annihilating ideal for f^s.)
893: (f is a polynomial of v1, ..., vn)
894: (<string> | <poly> f, s, v1, ..., vn ; <poly> L1, ..., Lm )
895: $Example: (x^3+y^3+z^3) [(s) (x) (y) (z)] genericAnn$
896: ]
897: ] putUsages ( genericAnn ) messagen-quiet
898: /fs.verbose 0 def
899: /genericAnn {
900: /arg2 set /arg1 set
901: [/in-genericAnn /f /vlist /s /vvv /nnn /rrr
902: /v1 /ops /ggg /ggg0
903: ] pushVariables
904: [(CurrentRingp) (KanGBmessage)] pushEnv
905: [
906: /f arg1 def /vlist arg2 def
907: f toString /f set
908: vlist { toString } map /vlist set
909: [(KanGBmessage) fs.verbose] system_variable
910: /s vlist 0 get def
911: /vvv (_u,_v,_t,) vlist rest { (,) 2 cat_n } map aload length /nnn set
912: s nnn 2 add cat_n def
913: fs.verbose { vvv message } { }ifelse
914: [vvv ring_of_differential_operators
915: [[(_u) 1 (_v) 1]] weight_vector 0] define_ring /rrr set
916:
917: [ (_u*_t). f . sub (_u*_v-1). ]
918: vlist rest { /v1 set
919: %%D-clean f . (D) v1 2 cat_n . 1 diff0 (_v*D_t). mul
920: f . @@@.Dsymbol v1 2 cat_n . 1 diff0 [(_v*) @@@.Dsymbol (_t)] cat . mul
921: @@@.Dsymbol v1 2 cat_n . add } map
922: join
923: /ops set
924: ops {[[(h). (1).]] replace } map /ops set
925: fs.verbose { ops message } { }ifelse
926: [ops] groebner_sugar 0 get /ggg0 set
927: fs.verbose { ggg0 message } { } ifelse
928: ggg0 [(_u) (_v)] eliminatev
929: %%D-clean { [(_t).] [ (D_t).] [s .] distraction
930: { [(_t).] [ [@@@.Dsymbol (_t)] cat .] [s .] distraction
931: [[s . << (0). s . sub (1). sub >>]] replace
932: } map /arg1 set
933: ] pop
934: popEnv
935: popVariables
936: arg1
937: } def
938:
939: %% Find differential equations for f^(m), r0 the minimal integral root.
940: [(annfs)
941: [( [ f v m r0] annfs g )
942: (It returns the annihilating ideal of f^m where r0 must be smaller)
943: (or equal to the minimal integral root of the b-function.)
944: (Or, it returns the annihilating ideal of f^r0, r0 and the b-function)
945: (where r0 is the minial integral root of b.)
946: (For the algorithm, see J. Pure and Applied Algebra 117&118(1997), 495--518.)
947: (Example 1: [(x^2+y^2+z^2+t^2) (x,y,z,t) -1 -2] annfs :: )
948: $ It returns the annihilating ideal of (x^2+y^2+z^2+t^2)^(-1).$
949: (Example 2: [(x^2+y^2+z^2+t^2) (x,y,z,t)] annfs :: )
950: $ It returns the annihilating ideal of f^r0 and [r0, b-function]$
951: $ where r0 is the minimal integral root of the b-function.$
952: (Example 3: [(x^2+y^2+z^2) (x,y,z) -1 -1] annfs :: )
953: (Example 4: [(x^3+y^3+z^3) (x,y,z)] annfs :: )
954: (Example 5: [((x1+x2+x3)(x1 x2 + x2 x3 + x1 x3) - t x1 x2 x3 ) )
955: ( (t,x1,x2,x3) -1 -2] annfs :: )
956: ( Note that the example 4 uses huge memory space.)
957: ]] putUsages
958: ( annfs ) messagen-quiet
959: /annfs.verbose fs.verbose def
960: /annfs.v [(x) (y) (z)] def
961: /annfs.s (_s) def
962: %% The first variable must be s.
963: /annfs {
964: /arg1 set
965: [/in-annfs /aa /typev /setarg /v /m /r0 /gg /ss /fs /gg2
966: /ans /vtmp /w2 /velim /bbb /rrr /r0
967: ] pushVariables
968: [(CurrentRingp) (KanGBmessage)] pushEnv
969: [
970:
971: /aa arg1 def
972: aa isArray { } { ( << array >> annfs) error } ifelse
973: /setarg 0 def
974: aa { tag } map /typev set
975: /r0 [ ] def
976: /m [ ] def
977: /v annfs.v def
978: aa 0 << aa 0 get toString >> put
979: typev [ StringP ] eq
980: { /f aa 0 get def
981: /setarg 1 def
982: } { } ifelse
983: typev [StringP StringP] eq
984: { /f aa 0 get def
985: /v [ aa 1 get to_records pop ] def
986: /setarg 1 def
987: } { } ifelse
988: typev [StringP ArrayP] eq
989: { /f aa 0 get def
990: /v aa 1 get def
991: /setarg 1 def
992: } { } ifelse
993: typev [StringP ArrayP IntegerP IntegerP] eq
994: { /f aa 0 get def
995: /v aa 1 get def
996: /m aa 2 get def
997: /r0 aa 3 get def
998: /setarg 1 def
999: } { } ifelse
1000: typev [StringP StringP IntegerP IntegerP] eq
1001: { /f aa 0 get def
1002: /v [ aa 1 get to_records pop ] def
1003: /m aa 2 get def
1004: /r0 aa 3 get def
1005: /setarg 1 def
1006: } { } ifelse
1007: setarg 1 eq { } { (annfs : wrong argument) error } ifelse
1008:
1009: [annfs.s] v join /v set
1010:
1011: /ss v 0 get def
1012: annfs.verbose {
1013: (f, v, s, f^{m}, m+r0 = ) messagen
1014: [ f (, ) v (, ) ss (, )
1015: (f^) m (,) m (+) r0 ] {messagen} map ( ) message
1016: } { } ifelse
1017:
1018: f v genericAnn /fs set
1019:
1020: annfs.verbose {
1021: (genericAnn is ) messagen fs message
1022: } { } ifelse
1023: [(KanGBmessage) annfs.verbose] system_variable
1024:
1025: m isArray {
1026: %% Now, let us find the b-function. /vtmp /w2 /velim /bbb /rrr /r0
1027: v rest { /vtmp set vtmp @@@.Dsymbol vtmp 2 cat_n } map /velim set
1028: velim { 1 } map /w2 set
1029: annfs.verbose { w2 message } { } ifelse
1030: [v from_records ring_of_differential_operators
1031: [w2] weight_vector 0] define_ring
1032: [ fs { toString . } map [ f toString . ] join ]
1033: groebner_sugar 0 get velim eliminatev 0 get /bbb set
1034: [[(s) annfs.s] from_records ring_of_polynomials 0] define_ring
1035: bbb toString . [[annfs.s . (s).]] replace /bbb set
1036: annfs.verbose { bbb message } { } ifelse
1037: bbb findIntegralRoots /rrr set
1038: rrr 0 get /r0 set %% minimal integral root.
1039: annfs.verbose { rrr message } { } ifelse
1040: fs 0 get (ring) dc ring_def
1041: fs { [[annfs.s . r0 toString .]] replace } map /ans set
1042: /ans [ans [r0 bbb]] def
1043: /annfs.label1 goto
1044: } { } ifelse
1045: m 0 ge {
1046: (annfs works only for getting annihilating ideal for f^(negative))
1047: error
1048: } { } ifelse
1049: r0 isArray {
1050: [(Need to compute the minimal root of b-function) nl
1051: (It has not been implemented.) ] cat
1052: error
1053: } { } ifelse
1054:
1055: [v from_records ring_of_differential_operators 0] define_ring
1056: fs {toString . dehomogenize [[ss . r0 (poly) dc]] replace}
1057: map /gg set
1058: annfs.verbose { gg message } { } ifelse
1059:
1060: [ [f . << m r0 sub >> npower ] gg join
1061: [(needBack) (needSyz)]] groebner_sugar 2 get /gg2 set
1062:
1063: gg2 { 0 get } map /ans set
1064: /ans ans { dup (0). eq {pop} { } ifelse } map def
1065:
1066: /annfs.label1
1067: /arg1 ans def
1068: ] pop
1069: popEnv
1070: popVariables
1071: arg1
1072: } def
1073:
1074: /genericAnnWithL.s (s) def
1075: /annfs.verify 0 def
1076: /genericAnnWithL {
1077: /arg1 set
1078: [/in-genericAnnWithL /aa /typev /setarg /v /m /r0 /gg /ss /fs /gg2
1079: /ans /vtmp /w2 /velim /bbb /rrr /r0 /myL /mygb /jj
1080: ] pushVariables
1081: [(CurrentRingp) (KanGBmessage) (Homogenize)] pushEnv
1082: [
1083:
1084: /aa arg1 def
1085: aa isArray { } { ( << array >> annfs) error } ifelse
1086: /setarg 0 def
1087: aa { tag } map /typev set
1088: /r0 [ ] def
1089: /m [ ] def
1090: /v annfs.v def
1091: aa 0 << aa 0 get toString >> put
1092: typev [ StringP ] eq
1093: { /f aa 0 get def
1094: /setarg 1 def
1095: } { } ifelse
1096: typev [StringP StringP] eq
1097: { /f aa 0 get def
1098: /v [ aa 1 get to_records pop ] def
1099: /setarg 1 def
1100: } { } ifelse
1101: typev [StringP ArrayP] eq
1102: { /f aa 0 get def
1103: /v aa 1 get def
1104: /setarg 1 def
1105: } { } ifelse
1106: setarg 1 eq { } { (genericAnnWithL : wrong argument) error } ifelse
1107:
1108: [genericAnnWithL.s] v join /v set
1109:
1110: /ss v 0 get def
1111: annfs.verbose {
1112: (f, v, s, f^{m}, m+r0 = ) messagen
1113: [ f (, ) v (, ) ss (, )
1114: (f^) m (,) m (+) r0 ] {messagen} map ( ) message
1115: } { } ifelse
1116:
1117: f v genericAnn /fs set
1118:
1119: annfs.verbose {
1120: (genericAnn is ) messagen fs message
1121: } { } ifelse
1122: [(KanGBmessage) annfs.verbose] system_variable
1123:
1124: m isArray {
1125: %% Now, let us find the b-function. /vtmp /w2 /velim /bbb /rrr /r0
1126: v rest { /vtmp set vtmp @@@.Dsymbol vtmp 2 cat_n } map /velim set
1127: velim { 1 } map /w2 set
1128: annfs.verbose { w2 message } { } ifelse
1129: [v from_records ring_of_differential_operators
1130: [w2] weight_vector 0] define_ring
1131:
1132: [ [ f toString . ] fs { toString . } map join [(needBack)]]
1133: groebner_sugar /mygb set
1134: mygb 0 get velim eliminatev 0 get /bbb set
1135: mygb 0 get bbb position /jj set
1136: mygb 1 get jj get 0 get /myL set
1137:
1138: annfs.verbose { bbb message } { } ifelse
1139:
1140: annfs.verify {
1141: (Verifying L f - b belongs to genericAnn(f)) message
1142: [(Homogenize) 0] system_variable
1143: << myL f . mul bbb sub >>
1144: [fs { toString . } map] groebner_sugar 0 get
1145: reduction 0 get message
1146: (Is it zero? Then it's fine.) message
1147: } { } ifelse
1148:
1149: /ans [bbb [myL fs] ] def
1150: /annfs.label1 goto
1151: } { } ifelse
1152:
1153: /annfs.label1
1154: /arg1 ans def
1155: ] pop
1156: popEnv
1157: popVariables
1158: arg1
1159: } def
1160:
1161:
1162: [(genericAnnWithL)
1163: [$[f v] genericAnnWithL [b [L I]]$
1164: $String f,v; poly b,L; array of poly I;$
1165: $f is a polynomial given by a string. v is the variables.$
1166: $ v must not contain names s, e.$
1167: $b is the b-function (Bernstein-Sato polynomial) for f and$
1168: $ L is the operator satisfying L f^{s+1} = b(s) f^s $
1169: $ I is the annihilating ideal of f^s.$
1170: $cf. bfunction, annfs, genericAnn.$
1171: $Example 1: [(x^2+y^2) (x,y)] genericAnnWithL ::$
1172: $Example 2: [(x^2+y^2+z^2) (x,y,z)] genericAnnWithL ::$
1173: $Example 3: [(x^3-y^2 z^2) (x,y,z)] genericAnnWithL ::$
1174: ]] putUsages
1.2 takayama 1175:
1176: /reduction*.noH 0 def
1177: /reduction* {
1178: /arg1 set
1179: [/in-reduction* /aa /typev /setarg /f /v
1180: /gg /wv /termorder /vec /ans /rr /mm /h /size /a0 /a3
1.3 takayama 1181: /opt
1.2 takayama 1182: ] pushVariables
1183: [(CurrentRingp) (KanGBmessage)] pushEnv
1184: [
1185:
1186: /aa arg1 def
1187: aa isArray { } { ( << array >> reduction*) error } ifelse
1188: /setarg 0 def
1189: /wv 0 def
1190: aa { tag } map /typev set
1191: typev [StringP ArrayP ArrayP] eq
1192: typev [ArrayP ArrayP ArrayP] eq or
1193: typev [PolyP ArrayP ArrayP] eq or
1194: { /h aa 0 get def
1195: /f aa 1 get def
1196: /v aa 2 get from_records def
1197: /setarg 1 def
1198: } { } ifelse
1199: typev [StringP ArrayP ArrayP ArrayP] eq
1200: typev [ArrayP ArrayP ArrayP ArrayP] eq or
1201: typev [PolyP ArrayP ArrayP ArrayP] eq or
1202: { /h aa 0 get def
1203: /f aa 1 get def
1204: /v aa 2 get from_records def
1205: /wv aa 3 get def
1206: /setarg 1 def
1207: } { } ifelse
1208:
1209: setarg { } { (reduction* : Argument mismatch) error } ifelse
1210:
1211: [(KanGBmessage) gb.verbose ] system_variable
1212:
1213: %%% Start of the preprocess
1214: f getRing /rr set
1215:
1216:
1217: rr tag 0 eq {
1218: %% Define our own ring
1219: v isInteger {
1220: (Error in reduction*: Specify variables) error
1221: } { } ifelse
1222: wv isInteger {
1223: [v ring_of_differential_operators
1224: 0] define_ring
1225: /termorder 1 def
1226: }{
1227: [v ring_of_differential_operators
1228: wv weight_vector
1229: 0] define_ring
1230: wv gb.isTermOrder /termorder set
1231: } ifelse
1232: } {
1233: %% Use the ring structre given by the input.
1234: v isInteger not {
1.7 takayama 1235: gb.warning {
1236: (Warning : the given ring definition is not used.) message
1237: } { } ifelse
1.2 takayama 1238: } { } ifelse
1239: rr ring_def
1240: /wv rr gb.getWeight def
1241: wv gb.isTermOrder /termorder set
1242: } ifelse
1243: %%% Enf of the preprocess
1244:
1245: f 0 get isArray {
1246: /size f 0 get length def
1247: f { { toString . } map } map /f set
1248: f fromVectors /f set
1249: }{
1250: /size -1 def
1251: f { toString . } map /f set
1252: } ifelse
1253:
1254: h isArray {
1255: h { toString . } map /h set
1256: [h] fromVectors 0 get /h set
1257: }{
1258: h toString . /h set
1259: } ifelse
1260: f { toString . } map /f set
1.3 takayama 1261: getOptions /opt set
1262: [(ReduceLowerTerms) 1] system_variable
1.2 takayama 1263: reduction*.noH {
1264: h f reduction-noH /ans set
1265: } {
1266: h f reduction /ans set
1267: } ifelse
1.3 takayama 1268: opt restoreOptions
1.2 takayama 1269: size -1 eq not {
1270: [size ans 0 get] toVectors /a0 set
1271: [size ans 3 get] toVectors /a3 set
1272: /ans [a0 ans 1 get ans 2 get a3] def
1273: } { } ifelse
1274: /arg1 ans def
1275: ] pop
1276: popEnv
1277: popVariables
1278: arg1
1279: } def
1280:
1281:
1282: [(reduction*)
1283: [([f base v] reduction* [h c0 syz input])
1284: ([f base v weight] reduction* [h c0 syz input])
1285: (reduction* is an user interface for reduction and reduction-noH.)
1286: (If reduction*.noH is one, then reduction-noH will be called.)
1287: (Example 1: [(x^2) [(x^2+y^2-4) (x y-1)] [(x) (y)]] reduction* )
1288: (Example 2: [[(1) (y^2-1)] [ [(0) (y-1)] [(1) (y+1)]] [(x) (y)]] reduction*)
1289: (Example 3: [(x^2) [(x^2+y^2-4) (x y-1)] [(x) (y)] [[(x) 10]] ] reduction* )
1290: ]] putUsages
1.5 takayama 1291:
1292:
1293:
1294: %% 2000, 6/7, at Sevilla, Hernando Colon
1295: %% macros that deal with homogenized inputs.
1296: %% Sample: [ [(h+x). (x^3).] [(x). (x).]] /ff set
1297: %% [(Homogenize_vec) 0] system_varialbe
1298: %% (grade) (grave1v) switch_function
1299: %% YA homogenization: [ [(h^3*(h+x)). (x^3).] [(h x). (x).]] /ff set
1300: %% 4+0 3+1 2+0 1+1
1301: /gb_h {
1302: /arg1 set
1303: [/in-gb_h /aa /typev /setarg /f /v
1304: /gg /wv /termorder /vec /ans /rr /mm
1305: /gb_h.opt
1306: ] pushVariables
1307: [(CurrentRingp) (KanGBmessage) (Homogenize_vec)] pushEnv
1308: [
1309:
1310: /aa arg1 def
1.6 takayama 1311: gb.verbose { (Getting in gb_h) message } { } ifelse
1.5 takayama 1312: aa isArray { } { ( << array >> gb_h) error } ifelse
1313: /setarg 0 def
1314: /wv 0 def
1315: aa { tag } map /typev set
1316: typev [ ArrayP ] eq
1317: { /f aa 0 get def
1318: /v gb.v def
1319: /setarg 1 def
1320: } { } ifelse
1321: typev [ArrayP StringP] eq
1322: { /f aa 0 get def
1323: /v aa 1 get def
1324: /setarg 1 def
1325: } { } ifelse
1326: typev [ArrayP ArrayP] eq
1327: { /f aa 0 get def
1328: /v aa 1 get from_records def
1329: /setarg 1 def
1330: } { } ifelse
1331: typev [ArrayP StringP ArrayP] eq
1332: { /f aa 0 get def
1333: /v aa 1 get def
1334: /wv aa 2 get def
1335: /setarg 1 def
1336: } { } ifelse
1337: typev [ArrayP ArrayP ArrayP] eq
1338: { /f aa 0 get def
1339: /v aa 1 get from_records def
1340: /wv aa 2 get def
1341: /setarg 1 def
1342: } { } ifelse
1343:
1344: setarg { } { (gb_h : Argument mismatch) error } ifelse
1345:
1346: [(KanGBmessage) gb.verbose ] system_variable
1347:
1348: %%% Start of the preprocess
1349: f getRing /rr set
1350: %% To the normal form : matrix expression.
1351: f gb.toMatrixOfString /f set
1352: /mm gb.itWasMatrix def
1353:
1354: rr tag 0 eq {
1355: %% Define our own ring
1356: v isInteger {
1357: (Error in gb_h: Specify variables) error
1358: } { } ifelse
1359: wv isInteger {
1360: [v ring_of_differential_operators
1361: 0] define_ring
1362: /termorder 1 def
1363: }{
1364: [v ring_of_differential_operators
1365: wv weight_vector
1366: 0] define_ring
1367: wv gb.isTermOrder /termorder set
1368: } ifelse
1369: } {
1370: %% Use the ring structre given by the input.
1371: v isInteger not {
1.7 takayama 1372: gb.warning {
1373: (Warning : the given ring definition is not used.) message
1374: } { } ifelse
1.5 takayama 1375: } { } ifelse
1376: rr ring_def
1377: /wv rr gb.getWeight def
1378: wv gb.isTermOrder /termorder set
1379: } ifelse
1380: getOptions /gb_h.opt set
1381: (grade) (module1v) switch_function
1.6 takayama 1382: [(Homogenize_vec) 0] system_variable
1.5 takayama 1383: %%% End of the preprocess
1384:
1385: gb.verbose { (gb.options = ) messagen gb.options message } { } ifelse
1386: termorder {
1387: f { {. } map } map /f set
1388: [f gb.options] groebner 0 get /gg set %% Do not use sugar.
1389: }{
1390: f { {. } map} map /f set
1391: f fromVectors /f set
1392: [f gb.options] groebner 0 get /gg set
1393: }ifelse
1394: wv isInteger {
1395: /ans [gg gg {init} map] def
1396: }{
1397: /ans [gg gg {wv 0 get weightv init} map] def
1398: }ifelse
1399:
1400: %% Postprocess : recover the matrix expression.
1401: mm {
1402: ans { /tmp set [mm tmp] toVectors } map
1403: /ans set
1404: }{ }
1405: ifelse
1406: gb_h.opt restoreOptions
1.6 takayama 1407: gb.verbose { (Getting out of gb_h) message } { } ifelse
1.5 takayama 1408: %%
1409:
1410: /arg1 ans def
1411: ] pop
1412: popEnv
1413: popVariables
1414: arg1
1415: } def
1416: (gb_h ) messagen-quiet
1417: [(gb_h)
1418: [(a gb_h b)
1419: (array a; array b;)
1420: (b : [g ii]; array g; array in; g is a Grobner basis of f)
1421: ( in the ring of homogenized differential operators.)
1422: ( The input must be homogenized properly.)
1423: ( Inproper homogenization may cause an infinite loop.)
1424: ( Each element of vectors must be homogenized. If you are using )
1425: ( non-term orders, all elements of vectors must have the same degree with)
1426: ( a proper degree shift vector.)
1427: $ ii is the initial ideal in case of w is given or <<a>> belongs$
1428: $ to a ring. In the other cases, it returns the initial monominal.$
1429: $ [(Homogenize_vec) 0] system_variable (grade) (module1v) switch_function$
1430: (a : [f ]; array f; f is a set of generators of an ideal in a ring.)
1431: (a : [f v]; array f; string v; v is the variables. )
1432: (a : [f v w]; array f; string v; array of array w; w is the weight matirx.)
1433: ( )
1434: $Example 1: [ [( (x Dx)^2 + (y Dy)^2 -h^4) ( x y Dx Dy -h^4)] (x,y) $
1435: $ [ [ (Dx) 1 ] ] ] gb_h pmat ; $
1436: $Example 2: [ [[(h+x) (x^3)] [(x) (x)]] (x)] gb_h pmat $
1437: $Example 3: [[ [(x^2) (y+x)] [(x+y) (y^3)] $
1438: $ [(2 x^2+x y) (y h^3 +x h^3 +x y^3)]] (x,y) $
1439: $ [ [ (x) -1 (y) -1] ] ] gb_h pmat ; $
1440: $ Infinite loop: see by [(DebugReductionRed) 1] system_variable$
1441: $Example 4: [[ [(x^2) (y+x)] [(x^2+y^2) (y)] $
1442: $ [(2 x^5+x y^4) (y h^3 +x h^3 +x y^3)]] (x,y) $
1443: $ [ [ (x) -1 (y) -1] ] ] gb_h pmat ; $
1444: $ This is fine because grade(v_1) = grade(v_2)+1 for all vectors. $
1445: ( )
1446: (cf. gb, groebner, syz_h. )
1447: ]] putUsages
1448:
1449: /syz_h {
1450: /arg1 set
1451: [/in-syz_h /aa /typev /setarg /f /v
1452: /gg /wv /termorder /vec /ans /ggall /vectorInput /vsize /gtmp /gtmp2
1453: /rr /mm
1454: /syz_h.opt
1455: ] pushVariables
1456: [(CurrentRingp) (KanGBmessage)] pushEnv
1457: [
1458:
1459: /aa arg1 def
1460: aa isArray { } { (<< array >> syz_h) error } ifelse
1461: /setarg 0 def
1462: /wv 0 def
1463: aa { tag } map /typev set
1464: typev [ ArrayP ] eq
1465: { /f aa 0 get def
1466: /v syz.v def
1467: /setarg 1 def
1468: } { } ifelse
1469: typev [ArrayP StringP] eq
1470: { /f aa 0 get def
1471: /v aa 1 get def
1472: /setarg 1 def
1473: } { } ifelse
1474: typev [ArrayP ArrayP] eq
1475: { /f aa 0 get def
1476: /v aa 1 get from_records def
1477: /setarg 1 def
1478: } { } ifelse
1479: typev [ArrayP StringP ArrayP] eq
1480: { /f aa 0 get def
1481: /v aa 1 get def
1482: /wv aa 2 get def
1483: /setarg 1 def
1484: } { } ifelse
1485: typev [ArrayP ArrayP ArrayP] eq
1486: { /f aa 0 get def
1487: /v aa 1 get from_records def
1488: /wv aa 2 get def
1489: /setarg 1 def
1490: } { } ifelse
1491:
1492: setarg { } { (syz_h : Argument mismatch) error } ifelse
1493:
1494: [(KanGBmessage) syz.verbose ] system_variable
1495:
1496:
1497:
1498: %%% Start of the preprocess
1499: f getRing /rr set
1500: %% To the normal form : matrix expression.
1501: f gb.toMatrixOfString /f set
1502: /mm gb.itWasMatrix def
1503: mm 0 gt {
1504: /vectorInput 1 def
1505: }{
1506: /vectorInput 1 def
1507: } ifelse
1508:
1509: rr tag 0 eq {
1510: %% Define our own ring
1511: v isInteger {
1512: (Error in syz_h: Specify variables) error
1513: } { } ifelse
1514: wv isInteger {
1515: [v ring_of_differential_operators
1516: 0] define_ring
1517: /termorder 1 def
1518: }{
1519: [v ring_of_differential_operators
1520: wv weight_vector
1521: 0] define_ring
1522: wv gb.isTermOrder /termorder set
1523: } ifelse
1524: }{
1525: %% Use the ring structre given by the input.
1526: v isInteger not {
1.7 takayama 1527: gb.warning {
1528: (Warning : the given ring definition is not used.) message
1529: } { } ifelse
1.5 takayama 1530: } { } ifelse
1531: rr ring_def
1532: /wv rr gb.getWeight def
1533: wv gb.isTermOrder /termorder set
1534: } ifelse
1535:
1536: getOptions /syz_h.opt set
1537: (grade) (module1v) switch_function
1538: [(Homogenize_vec) 0] system_variable
1539: %%% End of the preprocess
1540:
1541: termorder {
1542: f { {. } map } map /f set
1543: [f [(needBack) (needSyz)]] groebner /ggall set %% Do not use sugar.
1544: ggall 2 get /gg set
1545: }{
1546: f { {. } map } map /f set
1547: [f [(needBack) (needSyz)]] groebner /ggall set
1548: ggall 2 get /gg set
1549: }ifelse
1550: vectorInput {
1551: /vsize f 0 get length def %% input vector size.
1552: /gtmp ggall 0 get def
1553: [vsize gtmp] toVectors /gtmp set
1554: ggall 0 gtmp put
1555: }{ } ifelse
1556:
1557: syz_h.opt restoreOptions
1558: %%
1559:
1560: /arg1 [gg ggall] def
1561: ] pop
1562: popEnv
1563: popVariables
1564: arg1
1565: } def
1566: (syz_h ) messagen-quiet
1567:
1568: [(syz_h)
1569: [(a syz_h [b c])
1570: (array a; array b; array c)
1571: (b is a set of generators of the syzygies of f in the ring of)
1572: (homogenized differential operators.)
1573: ( The input must be homogenized properly.)
1574: ( Inproper homogenization may cause an infinite loop.)
1575: ( Each element of vectors must be homogenized. If you are using )
1576: ( non-term orders, all elements of vectors must have the same degree with)
1577: ( a proper degree shift vector.)
1578: (c = [gb, backward transformation, syzygy without dehomogenization].)
1579: (See gb_h.)
1580: $ [(Homogenize_vec) 0] system_variable (grade) (module1v) switch_function$
1581: (a : [f ]; array f; f is a set of generators of an ideal in a ring.)
1582: (a : [f v]; array f; string v; v is the variables.)
1583: (a : [f v w]; array f; string v; array of array w; w is the weight matirx.)
1584: $Example 1: [ [( (x Dx)^2 + (y Dy)^2 -h^4) ( x y Dx Dy -h^4)] (x,y) $
1585: $ [ [ (Dx) 1 ] ] ] syz_h pmat ; $
1586: $Example 2: [ [[(h+x) (x^3)] [(x) (x)]] (x)] syz_h pmat $
1587: $Example 3: [[ [(x^2) (y+x)] [(x+y) (y^3)] $
1588: $ [(2 x^2+x y) (y h^3 +x h^3 +x y^3)]] (x,y) $
1589: $ [ [ (x) -1 (y) -1] ] ] syz_h pmat ; $
1590: $ Infinite loop: see by [(DebugReductionRed) 1] system_variable$
1591: $Example 4: [[ [(x^2) (y+x)] [(x^2+y^2) (y)] $
1592: $ [(2 x^5+x y^4) (y h^3 +x h^3 +x y^3)]] (x,y) $
1593: $ [ [ (x) -1 (y) -1] ] ] syz_h pmat ; $
1594: $ This is fine because grade(v_1) = grade(v_2)+1 for all vectors. $
1595: $Example 5: [ [ [(0) (0)] [(0) (0)] [(x) (y)]] $
1596: $ [(x) (y)]] syz pmat ;$
1597: ]] putUsages
1598:
1599:
1600: /isSameIdeal {
1601: /arg1 set
1602: [/in-isSameIdeal /aa /ii /jj /iigg /jjgg /vv /ans /k /n /f] pushVariables
1603: [(CurrentRingp)] pushEnv
1604: [
1605: /aa arg1 def
1606: %% comparison of hilbert series has not yet been implemented.
1607: aa length 3 eq { }
1608: { ([ii jj vv] isSameIdeal) error } ifelse
1.6 takayama 1609: gb.verbose { (Getting in isSameIdeal) message } { } ifelse
1.5 takayama 1610: /ii aa 0 get def
1611: /jj aa 1 get def
1612: /vv aa 2 get def
1613: ii length 0 eq jj length 0 eq and
1614: { /ans 1 def /LLL.isSame goto } { } ifelse
1615: [ii vv] gb /iigg set
1616: [jj vv] gb /jjgg set
1617:
1618: iigg getRing ring_def
1619:
1620: /ans 1 def
1621: iigg 0 get { [ (toe_) 3 -1 roll ] gbext } map
1622: /iigg set
1623: jjgg 0 get { [ (toe_) 3 -1 roll ] gbext } map
1624: /jjgg set
1625:
1626: gb.verbose { ( ii < jj ?) messagen } { } ifelse
1627: iigg length /n set
1628: 0 1 n 1 sub {
1629: /k set
1630: iigg k get
1631: jjgg reduction-noH 0 get
1632: (0). eq not { /ans 0 def /LLL.isSame goto} { } ifelse
1633: gb.verbose { (o) messagen } { } ifelse
1634: } for
1635: gb.verbose { ( jj < ii ?) messagen } { } ifelse
1636: jjgg length /n set
1637: 0 1 n 1 sub {
1638: /k set
1639: jjgg k get
1640: iigg reduction-noH 0 get
1641: (0). eq not { /ans 0 def /LLL.isSame goto} { } ifelse
1642: gb.verbose { (o) messagen } { } ifelse
1643: } for
1644: /LLL.isSame
1645: gb.verbose { ( Done) message } { } ifelse
1646: /arg1 ans def
1647: ] pop
1648: popEnv
1649: popVariables
1650: arg1
1651: } def
1652: (isSameIdeal ) messagen-quiet
1653:
1654: [(isSameIdeal)
1655: [([ii jj vv] isSameIdeal bool)
1656: (ii, jj : ideal, vv : variables)
1657: (Note that ii and jj will be dehomogenized and compared in the ring)
1658: (of differential operators. cf. isSameIdeal_h)
1659: $Example 1: [ [(x^3) (y^2)] [(x^2+y) (y)] (x,y)] isSameIdeal $
1660: $Example 2: [ [[(x^3) (0)] [(y^2) (1)]] $
1661: $ [[(x^3+y^2) (1)] [(y^2) (1)]] (x,y)] isSameIdeal $
1662: ]] putUsages
1663:
1664: /isSameIdeal_h {
1665: /arg1 set
1.6 takayama 1666: [/in-isSameIdeal_h /aa /ii /jj /iigg /jjgg /vv /ans /k /n /f
1667: /isSameIdeal_h.opt
1668: ] pushVariables
1669: [(CurrentRingp) (Homogenize_vec)] pushEnv
1.5 takayama 1670: [
1671: /aa arg1 def
1.6 takayama 1672: gb.verbose { (Getting in isSameIdeal_h) message } { } ifelse
1.5 takayama 1673: %% comparison of hilbert series has not yet been implemented.
1674: aa length 3 eq { }
1675: { ([ii jj vv] isSameIdeal_h) error } ifelse
1676: /ii aa 0 get def
1677: /jj aa 1 get def
1678: /vv aa 2 get def
1679: ii length 0 eq jj length 0 eq and
1680: { /ans 1 def /LLL.isSame_h goto } { } ifelse
1681:
1682: [ii vv] gb_h /iigg set
1683: [jj vv] gb_h /jjgg set
1684:
1685: iigg getRing ring_def
1686:
1.6 takayama 1687: getOptions /isSameIdeal_h.opt set
1688: (grade) (module1v) switch_function
1689: [(Homogenize_vec) 0] system_variable
1.5 takayama 1690: /ans 1 def
1691: iigg 0 get { [ (toe_) 3 -1 roll ] gbext } map
1692: /iigg set
1693: jjgg 0 get { [ (toe_) 3 -1 roll ] gbext } map
1694: /jjgg set
1695:
1.8 ! takayama 1696: gb.verbose { (Comparing) message iigg message (and) message jjgg message }
! 1697: { } ifelse
1.5 takayama 1698: gb.verbose { ( ii < jj ?) messagen } { } ifelse
1699: iigg length /n set
1700: 0 1 n 1 sub {
1701: /k set
1702: iigg k get
1703: jjgg reduction 0 get
1704: (0). eq not { /ans 0 def /LLL.isSame_h goto} { } ifelse
1705: gb.verbose { (o) messagen } { } ifelse
1706: } for
1707: gb.verbose { ( jj < ii ?) messagen } { } ifelse
1708: jjgg length /n set
1709: 0 1 n 1 sub {
1710: /k set
1711: jjgg k get
1712: iigg reduction 0 get
1713: (0). eq not { /ans 0 def /LLL.isSame_h goto} { } ifelse
1714: gb.verbose { (o) messagen } { } ifelse
1715: } for
1716: /LLL.isSame_h
1717: gb.verbose { ( Done) message } { } ifelse
1.6 takayama 1718: isSameIdeal_h.opt restoreOptions
1.5 takayama 1719: /arg1 ans def
1720: ] pop
1721: popEnv
1722: popVariables
1723: arg1
1724: } def
1725: (isSameIdeal_h ) messagen-quiet
1726:
1727: [(isSameIdeal_h)
1728: [([ii jj vv] isSameIdeal_h bool)
1729: (ii, jj : ideal, vv : variables)
1730: (Note that ii and jj will be compared in the ring)
1731: (of homogenized differential operators. Each element of the vector must be)
1732: (homogenized.)
1733: $Example 1: [ [(x Dx - h^2) (Dx^2)] [(Dx^3) (x Dx-h^2)] (x)] isSameIdeal_h $
1734: $Example 2: [ [[(x Dx -h^2) (0)] [(Dx^2) (1)]] $
1735: $ [[(x Dx -h^2) (0)] [(Dx^2) (1)] [(Dx^3) (Dx)]] (x,y)] isSameIdeal_h $
1736: ]] putUsages
1737:
1738:
1.1 maekawa 1739:
1740: ( ) message-quiet ;
1741:
1742:
1743:
1744:
1745:
1746:
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