Annotation of OpenXM/src/k097/lib/minimal/minimal.k, Revision 1.15
1.15 ! takayama 1: /* $OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v 1.14 2000/06/09 08:04:54 takayama Exp $ */
1.1 takayama 2: #define DEBUG 1
3: /* #define ORDINARY 1 */
1.4 takayama 4: /* If you run this program on openxm version 1.1.2 (FreeBSD),
5: make a symbolic link by the command
6: ln -s /usr/bin/cpp /lib/cpp
7: */
1.6 takayama 8: #define OFFSET 0
1.14 takayama 9: #define TOTAL_STRATEGY 1
1.6 takayama 10: /* #define OFFSET 20*/
1.1 takayama 11: /* Test sequences.
12: Use load["minimal.k"];;
13:
14: a=Sminimal(v);
15: b=a[0];
16: b[1]*b[0]:
17: b[2]*b[1]:
18:
19: a = test0();
20: b = a[0];
21: b[1]*b[0]:
22: b[2]*b[1]:
23: a = Sminimal(b[0]);
24:
25: a = test1();
26: b=a[0];
27: b[1]*b[0]:
28: b[2]*b[1]:
29:
30: */
31:
32:
33: load("cohom.k");
34: def load_tower() {
35: if (Boundp("k0-tower.sm1.loaded")) {
36: }else{
37: sm1(" [(parse) (k0-tower.sm1) pushfile ] extension ");
38: sm1(" /k0-tower.sm1.loaded 1 def ");
39: }
1.7 takayama 40: sm1(" oxNoX ");
1.1 takayama 41: }
42: load_tower();
43: SonAutoReduce = true;
44: def Factor(f) {
45: sm1(f, " fctr /FunctionValue set");
46: }
47: def Reverse(f) {
48: sm1(f," reverse /FunctionValue set");
49: }
50: def Sgroebner(f) {
51: sm1(" [f] groebner /FunctionValue set");
52: }
53: def test0() {
54: local f;
55: Sweyl("x,y,z");
56: f = [x^2+y^2+z^2, x*y+x*z+y*z, x*z^2+y*z^2, y^3-x^2*z - x*y*z+y*z^2,
57: -y^2*z^2 + x*z^3 + y*z^3, -z^4];
58: frame=SresolutionFrame(f);
59: Println(frame);
60: /* return(frame); */
61: return(SlaScala(f));
62: }
63: def test1() {
64: local f;
65: Sweyl("x,y,z");
66: f = [x^2+y^2+z^2, x*y+x*z+y*z, x*z^2+y*z^2, y^3-x^2*z - x*y*z+y*z^2,
67: -y^2*z^2 + x*z^3 + y*z^3, -z^4];
68: return(Sminimal(f));
69: }
70:
71:
72:
73: def Sweyl(v,w) {
74: /* extern WeightOfSweyl ; */
75: local ww,i,n;
76: if(Length(Arglist) == 1) {
77: sm1(" [v s_ring_of_differential_operators 0 [(schreyer) 1]] define_ring ");
78: sm1(" define_ring_variables ");
79:
80: sm1(" [ v to_records pop ] /ww set ");
81: n = Length(ww);
82: WeightOfSweyl = NewArray(n*4);
83: for (i=0; i< n; i++) {
84: WeightOfSweyl[2*i] = ww[i];
85: WeightOfSweyl[2*i+1] = 1;
86: }
87: for (i=0; i< n; i++) {
88: WeightOfSweyl[2*n+2*i] = AddString(["D",ww[i]]);
89: WeightOfSweyl[2*n+2*i+1] = 1;
90: }
91:
92: }else{
93: sm1(" [v s_ring_of_differential_operators w s_weight_vector 0 [(schreyer) 1]] define_ring ");
94: sm1(" define_ring_variables ");
95: WeightOfSweyl = w[0];
96: }
97: }
98:
99:
100: def Spoly(f) {
101: sm1(f, " toString tparse /FunctionValue set ");
102: }
103:
104: def SreplaceZeroByZeroPoly(f) {
105: if (IsArray(f)) {
106: return(Map(f,"SreplaceZeroByZeroPoly"));
107: }else{
108: if (IsInteger(f)) {
109: return(Poly(ToString(f)));
110: }else{
111: return(f);
112: }
113: }
114: }
115: def Shomogenize(f) {
116: f = SreplaceZeroByZeroPoly(f);
117: if (IsArray(f)) {
118: sm1(f," sHomogenize2 /FunctionValue set ");
119: /* sm1(f," {sHomogenize2} map /FunctionValue set "); */
120: /* Is it correct? Double check.*/
121: }else{
122: sm1(f, " sHomogenize /FunctionValue set ");
123: }
124: }
125:
126: def StoTower() {
127: sm1(" [(AvoidTheSameRing)] pushEnv [ [(AvoidTheSameRing) 0] system_variable (mmLarger) (tower) switch_function ] pop popEnv ");
128: }
129:
130: def SsetTower(tower) {
131: sm1(" [(AvoidTheSameRing)] pushEnv
132: [ [(AvoidTheSameRing) 0] system_variable
133: [(gbListTower) tower (list) dc] system_variable
134: ] pop popEnv ");
1.14 takayama 135: /* sm1("(hoge) message show_ring "); */
1.1 takayama 136: }
137:
138: def SresolutionFrameWithTower(g,opt) {
139: local gbTower, ans, ff, count, startingGB, opts, skelton,withSkel, autof,
1.15 ! takayama 140: gbasis, nohomog;
! 141: nohomog = false;
! 142: count = -1;
1.1 takayama 143: if (Length(Arglist) >= 2) {
1.15 ! takayama 144: if (IsInteger(opt)) {
! 145: count = opt;
! 146: }else if (IsString(opt)) {
! 147: if (opt == "homogenized") {
! 148: nohomog = true;
! 149: }else{
! 150: Println("Warning: unknown option");
! 151: Println(opt);
! 152: }
! 153: }
1.1 takayama 154: }else{
155: count = -1;
156: }
157:
158: sm1(" setupEnvForResolution ");
159: /* If I do not put this macro, homogenization
160: make a strange behavior. For example,
161: [(2*x*Dx + 3*y*Dy+6) (0)] homogenize returns
162: [(2*x*Dx*h + 3*y*Dy*h+6*h^3) (0)].
163: 4/19, 2000.
164: */
165:
166: sm1(" (mmLarger) (matrix) switch_function ");
1.15 ! takayama 167: if (! nohomog) {
! 168: Println("Automatic homogenization.");
! 169: g = Map(g,"Shomogenize");
! 170: }else{
! 171: Println("No automatic homogenization.");
! 172: }
1.1 takayama 173: if (SonAutoReduce) {
174: sm1("[ (AutoReduce) ] system_variable /autof set ");
175: sm1("[ (AutoReduce) 1 ] system_variable ");
176: }
177: gbasis = Sgroebner(g);
178: g = gbasis[0];
179: if (SonAutoReduce) {
180: sm1("[ (AutoReduce) autof] system_variable ");
181: }
182:
183: g = Init(g);
184:
185: /* sm1(" setupEnvForResolution-sugar "); */
186: /* -sugar is fine? */
187: sm1(" setupEnvForResolution ");
188:
189: Println(g);
190: startingGB = g;
191: /* ans = [ SzeroMap(g) ]; It has not been implemented. see resol1.withZeroMap */
192: ans = [ ];
193: gbTower = [ ];
194: skelton = [ ];
195: while (true) {
196: /* sm1(g," res0Frame /ff set "); */
197: withSkel = Sres0FrameWithSkelton(g);
198: ff = withSkel[0];
199: ans = Append(ans, ff[0]);
200: gbTower = Join([ ff[1] ], gbTower);
201: skelton = Join([ withSkel[1] ], skelton);
202: g = ff[0];
203: if (Length(g) == 0) break;
204: SsetTower( gbTower );
205: if (count == 0) break;
206: count = count - 1;
207: }
208: return([ans,Reverse(gbTower),Join([ [ ] ], Reverse(skelton)),gbasis]);
209: }
210: HelpAdd(["SresolutionFrameWithTower",
211: ["It returs [resolution of the initial, gbTower, skelton, gbasis]",
1.15 ! takayama 212: "option: \"homogenized\" (no automatic homogenization) ",
1.1 takayama 213: "Example: Sweyl(\"x,y\");",
214: " a=SresolutionFrameWithTower([x^3,x*y,y^3-1]);"]]);
215:
216: def SresolutionFrame(f,opt) {
217: local ans;
1.15 ! takayama 218: ans = SresolutionFrameWithTower(f,opt);
1.1 takayama 219: return(ans[0]);
220: }
221: /* ---------------------------- */
222: def ToGradedPolySet(g) {
223: sm1(g," (gradedPolySet) dc /FunctionValue set ");
224: }
225:
226: def NewPolynomialVector(size) {
227: sm1(size," (integer) dc newPolyVector /FunctionValue set ");
228: }
229:
230: def SturnOffHomogenization() {
231: sm1("
232: [(Homogenize)] system_variable 1 eq
233: { (Warning: Homogenization and ReduceLowerTerms options are automatically turned off.) message
234: [(Homogenize) 0] system_variable
235: [(ReduceLowerTerms) 0] system_variable
236: } { } ifelse
237: ");
238: }
239: def SturnOnHomogenization() {
240: sm1("
241: [(Homogenize)] system_variable 0 eq
242: { (Warning: Homogenization and ReduceLowerTerms options are automatically turned ON.) message
243: [(Homogenize) 1] system_variable
244: [(ReduceLowerTerms) 1] system_variable
245: } { } ifelse
246: ");
247: }
248:
249: def SschreyerSkelton(g) {
250: sm1(" [(schreyerSkelton) g] gbext /FunctionValue set ");
251: }
252: def Stoes(g) {
253: if (IsArray(g)) {
254: sm1(g," {toes} map /FunctionValue set ");
255: }else{
256: sm1(g," toes /FunctionValue set ");
257: }
258: }
259: def Stoes_vec(g) {
260: sm1(g," toes /FunctionValue set ");
261: }
262:
263: def Sres0Frame(g) {
264: local ans;
265: ans = Sres0FrameWithSkelton(g);
266: return(ans[0]);
267: }
268: def Sres0FrameWithSkelton(g) {
269: local t_syz, nexttower, m, t_gb, skel, betti,
270: gg, k, i, j, pair, tmp, si, sj, grG, syzAll, gLength;
271:
272: SturnOffHomogenization();
273:
274: g = Stoes(g);
275: skel = SschreyerSkelton(g);
276: /* Print("Skelton is ");
277: sm1_pmat(skel); */
278: betti = Length(skel);
279:
280: gLength = Length(g);
281: grG = ToGradedPolySet(g);
282: syzAll = NewPolynomialVector(betti);
283: for (k=0; k<betti; k++) {
284: pair = skel[k];
285: i = pair[0,0];
286: j = pair[0,1];
287: si = pair[1,0];
288: sj = pair[1,1];
289: /* si g[i] + sj g[j] + \sum tmp[2][k] g[k] = 0 in res0 */
290: Print(".");
291:
292: t_syz = NewPolynomialVector(gLength);
293: t_syz[i] = si;
294: t_syz[j] = sj;
295: syzAll[k] = t_syz;
296: }
297: t_syz = syzAll;
298: Print("Done. betti="); Println(betti);
299: /* Println(g); g is in a format such as
300: [e_*x^2 , e_*x*y , 2*x*Dx*h , ...]
301: [e_*x^2 , e_*x*y , 2*x*Dx*h , ...]
302: [y-es*x , 3*es^4*y*Dy-es^5*x , 3*es^5*y*Dy-es^6*x , ...]
303: [3*es^3*y*Dy-es^5*x ]
304: */
305: nexttower = Init(g);
306: SturnOnHomogenization();
307: return([[t_syz, nexttower],skel]);
308: }
309:
310:
311: def StotalDegree(f) {
1.14 takayama 312: local d0;
313: sm1(" [(grade) f] gbext (universalNumber) dc /d0 set ");
314: /* Print("degree of "); Print(f); Print(" is "); Println(d0); */
315: return(d0);
1.1 takayama 316: }
317:
318: /* Sord_w(x^2*Dx*Dy,[x,-1,Dx,1]); */
319: def Sord_w(f,w) {
320: local neww,i,n;
321: n = Length(w);
322: neww = NewArray(n);
323: for (i=0; i<n; i=i+2) {
324: neww[i] = ToString(w[i]);
325: }
326: for (i=1; i<n; i=i+2) {
327: neww[i] = IntegerToSm1Integer(w[i]);
328: }
329: sm1(" f neww ord_w (universalNumber) dc /FunctionValue set ");
330: }
331:
332:
333: /* This is not satisfactory. */
334: def SinitOfArray(f) {
335: local p,pos,top;
336: if (IsArray(f)) {
337: sm1(f," toes init /p set ");
338: sm1(p," (es). degree (universalNumber) dc /pos set ");
339: return([Init(f[pos]),pos]);
340: } else {
341: return(Init(f));
342: }
343: }
344:
345: def test_SinitOfArray() {
346: local f, frame,p,tower,i,j,k;
347: Sweyl("x,y,z");
348: f = [x^2+y^2+z^2, x*y+x*z+y*z, x*z^2+y*z^2, y^3-x^2*z - x*y*z+y*z^2,
349: -y^2*z^2 + x*z^3 + y*z^3, -z^4];
350: p=SresolutionFrameWithTower(f);
351: sm1_pmat(p);
352: sm1_pmat(SgenerateTable(p[1]));
353: return(p);
354: frame = p[0];
355: sm1_pmat(p[1]);
356: sm1_pmat(frame);
357: sm1_pmat(Map(frame[0],"SinitOfArray"));
358: sm1_pmat(Map(frame[1],"SinitOfArray"));
359: return(p);
360: }
361:
362: /* f is assumed to be a monomial with toes. */
363: def Sdegree(f,tower,level) {
1.6 takayama 364: local i,ww, wd;
365: /* extern WeightOfSweyl; */
366: ww = WeightOfSweyl;
1.5 takayama 367: f = Init(f);
1.1 takayama 368: if (level <= 1) return(StotalDegree(f));
369: i = Degree(f,es);
1.6 takayama 370: #ifdef TOTAL_STRATEGY
371: return(StotalDegree(f)+Sdegree(tower[level-2,i],tower,level-1));
372: #endif
373: /* Strategy must be compatible with ordering. */
374: /* Weight vector must be non-negative, too. */
375: /* See Sdegree, SgenerateTable, reductionTable. */
376: wd = Sord_w(f,ww);
377: return(wd+Sdegree(tower[level-2,i],tower,level-1));
378:
1.1 takayama 379: }
380:
381: def SgenerateTable(tower) {
382: local height, n,i,j, ans, ans_at_each_floor;
383: height = Length(tower);
384: ans = NewArray(height);
385: for (i=0; i<height; i++) {
386: n = Length(tower[i]);
387: ans_at_each_floor=NewArray(n);
388: for (j=0; j<n; j++) {
1.6 takayama 389: ans_at_each_floor[j] = Sdegree(tower[i,j],tower,i+1)-(i+1)
390: + OFFSET;
1.1 takayama 391: /* Println([i,j,ans_at_each_floor[j]]); */
392: }
393: ans[i] = ans_at_each_floor;
394: }
395: return(ans);
396: }
397: Sweyl("x,y,z");
398: v=[[2*x*Dx + 3*y*Dy+6, 0],
399: [3*x^2*Dy + 2*y*Dx, 0],
400: [0, x^2+y^2],
401: [0, x*y]];
402: /* SresolutionFrameWithTower(v); */
403:
404: def SnewArrayOfFormat(p) {
405: if (IsArray(p)) {
406: return(Map(p,"SnewArrayOfFormat"));
407: }else{
408: return(null);
409: }
410: }
1.4 takayama 411: def ScopyArray(a) {
412: local n, i,ans;
413: n = Length(a);
414: ans = NewArray(n);
415: for (i=0; i<n; i++) {
416: ans[i] = a[i];
417: }
418: return(ans);
419: }
1.1 takayama 420: def SminOfStrategy(a) {
421: local n,i,ans,tt;
422: ans = 100000; /* very big number */
423: if (IsArray(a)) {
424: n = Length(a);
425: for (i=0; i<n; i++) {
426: if (IsArray(a[i])) {
427: tt = SminOfStrategy(a[i]);
428: if (tt < ans) ans = tt;
429: }else{
430: if (a[i] < ans) ans = a[i];
431: }
432: }
433: }else{
434: if (a < ans) ans = a;
435: }
436: return(ans);
437: }
438: def SmaxOfStrategy(a) {
439: local n,i,ans,tt;
440: ans = -100000; /* very small number */
441: if (IsArray(a)) {
442: n = Length(a);
443: for (i=0; i<n; i++) {
444: if (IsArray(a[i])) {
445: tt = SmaxOfStrategy(a[i]);
446: if (tt > ans) ans = tt;
447: }else{
448: if (a[i] > ans) ans = a[i];
449: }
450: }
451: }else{
452: if (a > ans) ans = a;
453: }
454: return(ans);
455: }
456:
457:
1.15 ! takayama 458: def SlaScala(g,opt) {
1.1 takayama 459: local rf, tower, reductionTable, skel, redundantTable, bases,
460: strategy, maxOfStrategy, height, level, n, i,
461: freeRes,place, f, reducer,pos, redundant_seq,bettiTable,freeResV,ww,
1.4 takayama 462: redundantTable_ordinary, redundant_seq_ordinary,
463: reductionTable_tmp;
1.1 takayama 464: /* extern WeightOfSweyl; */
465: ww = WeightOfSweyl;
1.6 takayama 466: Print("WeightOfSweyl="); Println(WeightOfSweyl);
1.15 ! takayama 467: rf = SresolutionFrameWithTower(g,opt);
1.14 takayama 468: Print("rf="); sm1_pmat(rf);
1.1 takayama 469: redundant_seq = 1; redundant_seq_ordinary = 1;
470: tower = rf[1];
471: reductionTable = SgenerateTable(tower);
472: skel = rf[2];
473: redundantTable = SnewArrayOfFormat(rf[1]);
474: redundantTable_ordinary = SnewArrayOfFormat(rf[1]);
475: reducer = SnewArrayOfFormat(rf[1]);
476: freeRes = SnewArrayOfFormat(rf[1]);
477: bettiTable = SsetBettiTable(rf[1],g);
478:
479: strategy = SminOfStrategy( reductionTable );
480: maxOfStrategy = SmaxOfStrategy( reductionTable );
481: height = Length(reductionTable);
482: while (strategy <= maxOfStrategy) {
483: for (level = 0; level < height; level++) {
484: n = Length(reductionTable[level]);
1.4 takayama 485: reductionTable_tmp = ScopyArray(reductionTable[level]);
486: while (SthereIs(reductionTable_tmp,strategy)) {
487: i = SnextI(reductionTable_tmp,strategy,redundantTable,
488: skel,level,freeRes);
489: Println([level,i]);
490: reductionTable_tmp[i] = -200000;
1.1 takayama 491: if (reductionTable[level,i] == strategy) {
492: Print("Processing "); Print([level,i]);
493: Print(" Strategy = "); Println(strategy);
494: if (level == 0) {
495: if (IsNull(redundantTable[level,i])) {
496: bases = freeRes[level];
497: /* Println(["At floor : GB=",i,bases,tower[0,i]]); */
498: pos = SwhereInGB(tower[0,i],rf[3,0]);
499: bases[i] = rf[3,0,pos];
500: redundantTable[level,i] = 0;
501: redundantTable_ordinary[level,i] = 0;
502: freeRes[level] = bases;
503: /* Println(["GB=",i,bases,tower[0,i]]); */
504: }
505: }else{ /* level >= 1 */
506: if (IsNull(redundantTable[level,i])) {
507: bases = freeRes[level];
508: f = SpairAndReduction(skel,level,i,freeRes,tower,ww);
509: if (f[0] != Poly("0")) {
510: place = f[3];
511: /* (level-1, place) is the place for f[0],
512: which is a newly obtained GB. */
513: #ifdef ORDINARY
514: redundantTable[level-1,place] = redundant_seq;
515: redundant_seq++;
516: #else
517: if (f[4] > f[5]) {
518: /* Zero in the gr-module */
519: Print("v-degree of [org,remainder] = ");
520: Println([f[4],f[5]]);
521: Print("[level,i] = "); Println([level,i]);
522: redundantTable[level-1,place] = 0;
523: }else{
524: redundantTable[level-1,place] = redundant_seq;
525: redundant_seq++;
526: }
527: #endif
528: redundantTable_ordinary[level-1,place]
529: =redundant_seq_ordinary;
530: redundant_seq_ordinary++;
531: bases[i] = SunitOfFormat(place,f[1])-f[1]; /* syzygy */
532: redundantTable[level,i] = 0;
533: redundantTable_ordinary[level,i] = 0;
534: /* i must be equal to f[2], I think. Double check. */
535: freeRes[level] = bases;
536: bases = freeRes[level-1];
537: bases[place] = f[0];
538: freeRes[level-1] = bases;
539: reducer[level-1,place] = f[1];
540: }else{
541: redundantTable[level,i] = 0;
542: bases = freeRes[level];
543: bases[i] = f[1]; /* Put the syzygy. */
544: freeRes[level] = bases;
545: }
546: }
547: } /* end of level >= 1 */
548: }
549: }
550: }
551: strategy++;
552: }
553: n = Length(freeRes);
554: freeResV = SnewArrayOfFormat(freeRes);
555: for (i=0; i<n; i++) {
556: bases = freeRes[i];
557: bases = Sbases_to_vec(bases,bettiTable[i]);
558: freeResV[i] = bases;
559: }
560: return([freeResV, redundantTable,reducer,bettiTable,redundantTable_ordinary]);
561: }
1.4 takayama 562:
563: def SthereIs(reductionTable_tmp,strategy) {
564: local n,i;
565: n = Length(reductionTable_tmp);
566: for (i=0; i<n; i++) {
567: if (reductionTable_tmp[i] == strategy) {
568: return(true);
569: }
570: }
571: return(false);
572: }
573:
574: def SnextI(reductionTable_tmp,strategy,redundantTable,
575: skel,level,freeRes)
576: {
577: local ii,n,p,myindex,i,j,bases;
578: n = Length(reductionTable_tmp);
579: if (level == 0) {
580: for (ii=0; ii<n; ii++) {
581: if (reductionTable_tmp[ii] == strategy) {
582: return(ii);
583: }
584: }
585: }else{
586: for (ii=0; ii<n; ii++) {
587: if (reductionTable_tmp[ii] == strategy) {
588: p = skel[level,ii];
589: myindex = p[0];
590: i = myindex[0]; j = myindex[1];
591: bases = freeRes[level-1];
592: if (IsNull(bases[i]) || IsNull(bases[j])) {
593:
594: }else{
595: return(ii);
596: }
597: }
598: }
599: }
1.5 takayama 600: Print("reductionTable_tmp=");
1.4 takayama 601: Println(reductionTable_tmp);
1.5 takayama 602: Println("See also reductionTable, strategy, level,i");
1.4 takayama 603: Error("SnextI: bases[i] or bases[j] is null for all combinations.");
604: }
605:
606:
1.1 takayama 607:
608: def SsetBettiTable(freeRes,g) {
609: local level,i, n,bases,ans;
610: ans = NewArray(Length(freeRes)+1);
611: n = Length(freeRes);
612: if (IsArray(g[0])) {
613: ans[0] = Length(g[0]);
614: }else{
615: ans[0] = 1;
616: }
617: for (level=0; level<n; level++) {
618: bases = freeRes[level];
619: if (IsArray(bases)) {
620: ans[level+1] = Length(bases);
621: }else{
622: ans[level+1] = 1;
623: }
624: }
625: return(ans);
626: }
627:
628: def SwhereInGB(f,tower) {
629: local i,n,p,q;
630: n = Length(tower);
631: for (i=0; i<n; i++) {
632: p = MonomialPart(tower[i]);
633: q = MonomialPart(f);
634: if (p == q) return(i);
635: }
636: Println([f,tower]);
637: Error("whereInGB : [f,myset]: f could not be found in the myset.");
638: }
639: def SunitOfFormat(pos,forms) {
640: local ans,i,n;
641: n = Length(forms);
642: ans = NewArray(n);
643: for (i=0; i<n; i++) {
644: if (i != pos) {
645: ans[i] = Poly("0");
646: }else{
647: ans[i] = Poly("1");
648: }
649: }
650: return(ans);
651: }
652:
653: def Error(s) {
654: sm1(" s error ");
655: }
656:
657: def IsNull(s) {
658: if (Stag(s) == 0) return(true);
659: else return(false);
660: }
661:
662: def StowerOf(tower,level) {
663: local ans,i;
664: ans = [ ];
665: if (level == 0) return([[]]);
666: for (i=0; i<level; i++) {
667: ans = Append(ans,tower[i]);
668: }
669: return(Reverse(ans));
670: }
671:
672: def Sspolynomial(f,g) {
673: if (IsArray(f)) {
674: f = Stoes_vec(f);
675: }
676: if (IsArray(g)) {
677: g = Stoes_vec(g);
678: }
679: sm1("f g spol /FunctionValue set");
680: }
681:
682: def MonomialPart(f) {
683: sm1(" [(lmonom) f] gbext /FunctionValue set ");
684: }
685:
1.14 takayama 686: /* WARNING:
687: When you use SwhereInTower, you have to change gbList
688: as below. Ofcourse, you should restrore the gbList
689: SsetTower(StowerOf(tower,level));
690: pos = SwhereInTower(syzHead,tower[level]);
691: */
1.1 takayama 692: def SwhereInTower(f,tower) {
693: local i,n,p,q;
694: if (f == Poly("0")) return(-1);
695: n = Length(tower);
696: for (i=0; i<n; i++) {
697: p = MonomialPart(tower[i]);
698: q = MonomialPart(f);
699: if (p == q) return(i);
700: }
701: Println([f,tower]);
702: Error("[f,tower]: f could not be found in the tower.");
703: }
704:
705: def Stag(f) {
706: sm1(f," tag (universalNumber) dc /FunctionValue set");
707: }
708:
709: def SpairAndReduction(skel,level,ii,freeRes,tower,ww) {
710: local i, j, myindex, p, bases, tower2, gi, gj,
711: si, sj, tmp, t_syz, pos, ans, ssp, syzHead,pos2,
712: vdeg,vdeg_reduced;
713: Println("SpairAndReduction:");
714:
715: if (level < 1) Error("level should be >= 1 in SpairAndReduction.");
716: p = skel[level,ii];
717: myindex = p[0];
718: i = myindex[0]; j = myindex[1];
719: bases = freeRes[level-1];
720: Println(["p and bases ",p,bases]);
721: if (IsNull(bases[i]) || IsNull(bases[j])) {
722: Println([level,i,j,bases[i],bases[j]]);
723: Error("level, i, j : bases[i], bases[j] must not be NULL.");
724: }
725:
726: tower2 = StowerOf(tower,level-1);
727: SsetTower(tower2);
1.14 takayama 728: Println(["level=",level]);
729: Println(["tower2=",tower2]);
1.1 takayama 730: /** sm1(" show_ring "); */
731:
732: gi = Stoes_vec(bases[i]);
733: gj = Stoes_vec(bases[j]);
734:
735: ssp = Sspolynomial(gi,gj);
736: si = ssp[0,0];
737: sj = ssp[0,1];
738: syzHead = si*es^i;
739: /* This will be the head term, I think. But, double check. */
740: Println([si*es^i,sj*es^j]);
741:
742: Print("[gi, gj] = "); Println([gi,gj]);
743: sm1(" [(Homogenize)] system_variable message ");
744: Print("Reduce the element "); Println(si*gi+sj*gj);
745: Print("by "); Println(bases);
746:
747: tmp = Sreduction(si*gi+sj*gj, bases);
748:
749: Print("result is "); Println(tmp);
750:
1.3 takayama 751: /* This is essential part for V-minimal resolution. */
752: /* vdeg = SvDegree(si*gi+sj*gj,tower,level-1,ww); */
753: vdeg = SvDegree(si*gi,tower,level-1,ww);
1.1 takayama 754: vdeg_reduced = SvDegree(tmp[0],tower,level-1,ww);
755: Print("vdegree of the original = "); Println(vdeg);
756: Print("vdegree of the remainder = "); Println(vdeg_reduced);
757:
758: t_syz = tmp[2];
759: si = si*tmp[1]+t_syz[i];
760: sj = sj*tmp[1]+t_syz[j];
761: t_syz[i] = si;
762: t_syz[j] = sj;
1.14 takayama 763:
764: SsetTower(StowerOf(tower,level));
1.1 takayama 765: pos = SwhereInTower(syzHead,tower[level]);
1.14 takayama 766:
767: SsetTower(StowerOf(tower,level-1));
1.1 takayama 768: pos2 = SwhereInTower(tmp[0],tower[level-1]);
769: ans = [tmp[0],t_syz,pos,pos2,vdeg,vdeg_reduced];
770: /* pos is the place to put syzygy at level. */
771: /* pos2 is the place to put a new GB at level-1. */
772: Println(ans);
773: return(ans);
774: }
775:
776: def Sreduction(f,myset) {
777: local n, indexTable, set2, i, j, tmp, t_syz;
778: n = Length(myset);
779: indexTable = NewArray(n);
780: set2 = [ ];
781: j = 0;
782: for (i=0; i<n; i++) {
783: if (IsNull(myset[i])) {
784: indexTable[i] = -1;
785: /* }else if (myset[i] == Poly("0")) {
786: indexTable[i] = -1; */
787: }else{
788: set2 = Append(set2,Stoes_vec(myset[i]));
789: indexTable[i] = j;
790: j++;
791: }
792: }
793: sm1(" f toes set2 (gradedPolySet) dc reduction /tmp set ");
794: t_syz = NewArray(n);
795: for (i=0; i<n; i++) {
796: if (indexTable[i] != -1) {
797: t_syz[i] = tmp[2, indexTable[i]];
798: }else{
799: t_syz[i] = Poly("0");
800: }
801: }
802: return([tmp[0],tmp[1],t_syz]);
803: }
804:
805: def Warning(s) {
806: Print("Warning: ");
807: Println(s);
808: }
809: def RingOf(f) {
810: local r;
811: if (IsPolynomial(f)) {
812: if (f != Poly("0")) {
813: sm1(f," (ring) dc /r set ");
814: }else{
815: sm1(" [(CurrentRingp)] system_variable /r set ");
816: }
817: }else{
818: Warning("RingOf(f): the argument f must be a polynomial. Return the current ring.");
819: sm1(" [(CurrentRingp)] system_variable /r set ");
820: }
821: return(r);
822: }
823:
824: def Sfrom_es(f,size) {
825: local c,ans, i, d, myes, myee, j,n,r,ans2;
826: if (Length(Arglist) < 2) size = -1;
827: if (IsArray(f)) return(f);
828: r = RingOf(f);
829: myes = PolyR("es",r);
830: myee = PolyR("e_",r);
831: if (Degree(f,myee) > 0 && size == -1) {
832: if (size == -1) {
833: sm1(f," (array) dc /ans set");
834: return(ans);
835: }
836: }
837:
838: /*
839: Coefficients(x^2-1,x):
840: [ [ 2 , 0 ] , [ 1 , -1 ] ]
841: */
842: if (Degree(f,myee) > 0) {
843: c = Coefficients(f,myee);
844: }else{
845: c = Coefficients(f,myes);
846: }
847: if (size < 0) {
848: size = c[0,0]+1;
849: }
850: ans = NewArray(size);
851: for (i=0; i<size; i++) {ans[i] = 0;}
852: n = Length(c[0]);
853: for (j=0; j<n; j++) {
854: d = c[0,j];
855: ans[d] = c[1,j];
856: }
857: return(ans);
858: }
859:
860: def Sbases_to_vec(bases,size) {
861: local n, giveSize, newbases,i;
862: /* bases = [1+es*x, [1,2,3*x]] */
863: if (Length(Arglist) > 1) {
864: giveSize = true;
865: }else{
866: giveSize = false;
867: }
868: n = Length(bases);
869: newbases = NewArray(n);
870: for (i=0; i<n; i++) {
871: if (giveSize) {
872: newbases[i] = Sfrom_es(bases[i], size);
873: }else{
874: newbases[i] = Sfrom_es(bases[i]);
875: }
876: }
877: return(newbases);
878: }
879:
1.14 takayama 880: HelpAdd(["Sminimal",
881: ["It constructs the V-minimal free resolution by LaScala-Stillman's algorithm",
1.15 ! takayama 882: "option: \"homogenized\" (no automatic homogenization ",
1.14 takayama 883: "Example: Sweyl(\"x,y\",[[\"x\",-1,\"y\",-1,\"Dx\",1,\"Dy\",1]]);",
884: " v=[[2*x*Dx + 3*y*Dy+6, 0],",
885: " [3*x^2*Dy + 2*y*Dx, 0],",
886: " [0, x^2+y^2],",
887: " [0, x*y]];",
888: " a=Sminimal(v);",
889: " Sweyl(\"x,y\",[[\"x\",-1,\"y\",-1,\"Dx\",1,\"Dy\",1]]);",
890: " b = ReParse(a[0]); sm1_pmat(b); ",
891: " IsExact_h(b,[x,y]):",
892: "Note: a[0] is the V-minimal resolution. a[3] is the Schreyer resolution."]]);
893:
1.15 ! takayama 894: def Sminimal(g,opt) {
1.1 takayama 895: local r, freeRes, redundantTable, reducer, maxLevel,
896: minRes, seq, maxSeq, level, betti, q, bases, dr,
1.14 takayama 897: betti_levelplus, newbases, i, j,qq, tminRes;
1.15 ! takayama 898: r = SlaScala(g,opt);
1.1 takayama 899: /* Should I turn off the tower?? */
900: freeRes = r[0];
901: redundantTable = r[1];
902: reducer = r[2];
903: minRes = SnewArrayOfFormat(freeRes);
904: seq = 0;
905: maxSeq = SgetMaxSeq(redundantTable);
906: maxLevel = Length(freeRes);
907: for (level = 0; level < maxLevel; level++) {
908: minRes[level] = freeRes[level];
909: }
910: seq=maxSeq+1;
911: while (seq > 1) {
912: seq--;
913: for (level = 0; level < maxLevel; level++) {
914: betti = Length(freeRes[level]);
915: for (q = 0; q<betti; q++) {
916: if (redundantTable[level,q] == seq) {
917: Print("[seq,level,q]="); Println([seq,level,q]);
918: if (level < maxLevel-1) {
919: bases = freeRes[level+1];
920: dr = reducer[level,q];
921: dr[q] = -1;
922: newbases = SnewArrayOfFormat(bases);
923: betti_levelplus = Length(bases);
924: /*
925: bases[i,j] ---> bases[i,j]+bases[i,q]*dr[j]
926: */
927: for (i=0; i<betti_levelplus; i++) {
928: newbases[i] = bases[i] + bases[i,q]*dr;
929: }
930: Println(["level, q =", level,q]);
931: Println("bases="); sm1_pmat(bases);
932: Println("dr="); sm1_pmat(dr);
933: Println("newbases="); sm1_pmat(newbases);
934: minRes[level+1] = newbases;
935: freeRes = minRes;
936: #ifdef DEBUG
937: for (qq=0; qq<betti; qq++) {
938: if ((redundantTable[level,qq] >= seq) &&
939: (redundantTable[level,qq] <= maxSeq)) {
940: for (i=0; i<betti_levelplus; i++) {
941: if (!IsZero(newbases[i,qq])) {
942: Println(["[i,qq]=",[i,qq]," is not zero in newbases."]);
943: Print("redundantTable ="); sm1_pmat(redundantTable[level]);
944: Error("Stop in Sminimal for debugging.");
945: }
946: }
947: }
948: }
949: #endif
950: }
951: }
952: }
953: }
954: }
1.14 takayama 955: tminRes = Stetris(minRes,redundantTable);
956: return([SpruneZeroRow(tminRes), tminRes,
1.3 takayama 957: [ minRes, redundantTable, reducer,r[3],r[4]],r[0]]);
1.1 takayama 958: /* r[4] is the redundantTable_ordinary */
1.3 takayama 959: /* r[0] is the freeResolution */
1.1 takayama 960: }
961:
962:
963: def IsZero(f) {
964: if (IsPolynomial(f)) {
965: return( f == Poly("0"));
966: }else if (IsInteger(f)) {
967: return( f == 0);
968: }else if (IsSm1Integer(f)) {
969: return( f == true );
970: }else if (IsDouble(f)) {
971: return( f == 0.0 );
972: }else if (IsRational(f)) {
973: return(IsZero(Denominator(f)));
974: }else{
975: Error("IsZero: cannot deal with this data type.");
976: }
977: }
978: def SgetMaxSeq(redundantTable) {
979: local level,i,n,ans, levelMax,bases;
980: levelMax = Length( redundantTable );
981: ans = 0;
982: for (level = 0; level < levelMax; level++) {
983: bases = redundantTable[level];
984: n = Length(bases);
985: for (i=0; i<n; i++) {
986: if (IsInteger( bases[i] )) {
987: if (bases[i] > ans) {
988: ans = bases[i];
989: }
990: }
991: }
992: }
993: return(ans);
994: }
995:
996: def Stetris(freeRes,redundantTable) {
997: local level, i, j, resLength, minRes,
998: bases, newbases, newbases2;
999: minRes = SnewArrayOfFormat(freeRes);
1000: resLength = Length( freeRes );
1001: for (level=0; level<resLength; level++) {
1002: bases = freeRes[level];
1003: newbases = SnewArrayOfFormat(bases);
1004: betti = Length(bases); j = 0;
1005: /* Delete rows */
1006: for (i=0; i<betti; i++) {
1007: if (redundantTable[level,i] < 1) {
1008: newbases[j] = bases[i];
1009: j++;
1010: }
1011: }
1012: bases = SfirstN(newbases,j);
1013: if (level > 0) {
1014: /* Delete columns */
1015: newbases = Transpose(bases);
1016: betti = Length(newbases); j = 0;
1017: newbases2 = SnewArrayOfFormat(newbases);
1018: for (i=0; i<betti; i++) {
1019: if (redundantTable[level-1,i] < 1) {
1020: newbases2[j] = newbases[i];
1021: j++;
1022: }
1023: }
1024: newbases = Transpose(SfirstN(newbases2,j));
1025: }else{
1026: newbases = bases;
1027: }
1028: Println(["level=", level]);
1029: sm1_pmat(bases);
1030: sm1_pmat(newbases);
1031:
1032: minRes[level] = newbases;
1033: }
1034: return(minRes);
1035: }
1036:
1037: def SfirstN(bases,k) {
1038: local ans,i;
1039: ans = NewArray(k);
1040: for (i=0; i<k; i++) {
1041: ans[i] = bases[i];
1042: }
1043: return(ans);
1044: }
1045:
1046:
1047: /* usage: tt is tower. ww is weight.
1048: a = SresolutionFrameWithTower(v);
1049: tt = a[1];
1050: ww = [x,1,y,1,Dx,1,Dy,1];
1051: SvDegree(x*es,tt,1,ww):
1052:
1053: In(17)=tt:
1054: [[2*x*Dx , e_*x^2 , e_*x*y , 3*x^2*Dy , e_*y^3 , 9*x*y*Dy^2 , 27*y^2*Dy^3 ] ,
1055: [es*y , 3*es^3*y*Dy , 3*es^5*y*Dy , 3*x*Dy , es^2*y^2 , 9*y*Dy^2 ] ,
1056: [3*es^3*y*Dy ] ]
1057: In(18)=SvDegree(x*es,tt,1,ww):
1058: 3
1059: In(19)=SvDegree(x*es^3,tt,1,ww):
1060: 4
1061: In(20)=SvDegree(x,tt,2,ww):
1062: 4
1063:
1064: */
1065: def SvDegree(f,tower,level,w) {
1066: local i,ans;
1067: if (IsZero(f)) return(null);
1.3 takayama 1068: f = Init(f);
1.1 takayama 1069: if (level <= 0) {
1070: return(Sord_w(f,w));
1071: }
1072: i = Degree(f,es);
1073: ans = Sord_w(f,w) +
1074: SvDegree(tower[level-1,i],tower,level-1,w);
1075: return(ans);
1076: }
1077:
1.2 takayama 1078: def Sannfs(f,v) {
1079: local f2;
1080: f2 = ToString(f);
1081: if (IsArray(v)) {
1082: v = Map(v,"ToString");
1083: }
1084: sm1(" [f2 v] annfs /FunctionValue set ");
1085: }
1086:
1087: /* Sannfs2("x^3-y^2"); */
1088: def Sannfs2(f) {
1089: local p,pp;
1090: p = Sannfs(f,"x,y");
1.6 takayama 1091: sm1(" p 0 get { [(x) (y) (Dx) (Dy)] laplace0 } map /p set ");
1.5 takayama 1092: /*
1093: Sweyl("x,y",[["x",1,"y",1,"Dx",1,"Dy",1,"h",1],
1094: ["x",-1,"y",-1,"Dx",1,"Dy",1]]); */
1.6 takayama 1095: /* Sweyl("x,y",[["x",1,"y",1,"Dx",1,"Dy",1,"h",1]]); */
1.10 takayama 1096:
1.6 takayama 1097: Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
1098: pp = Map(p,"Spoly");
1099: return(Sminimal_v(pp));
1100: /* return(Sminimal(pp)); */
1101: }
1102:
1.10 takayama 1103: HelpAdd(["Sannfs2",
1104: ["Sannfs2(f) constructs the V-minimal free resolution for the weight (-1,1)",
1105: "of the Laplace transform of the annihilating ideal of the polynomial f in x,y.",
1106: "See also Sminimal_v, Sannfs3.",
1107: "Example: a=Sannfs2(\"x^3-y^2\");",
1108: " b=a[0]; sm1_pmat(b);",
1109: " b[1]*b[0]:",
1110: "Example: a=Sannfs2(\"x*y*(x-y)*(x+y)\");",
1111: " b=a[0]; sm1_pmat(b);",
1112: " b[1]*b[0]:"
1113: ]]);
1114:
1.6 takayama 1115: /* Do not forget to turn on TOTAL_STRATEGY */
1116: def Sannfs2_laScala(f) {
1117: local p,pp;
1118: p = Sannfs(f,"x,y");
1119: /* Do not make laplace transform.
1120: sm1(" p 0 get { [(x) (y) (Dx) (Dy)] laplace0 } map /p set ");
1121: p = [p];
1122: */
1123: Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
1.2 takayama 1124: pp = Map(p[0],"Spoly");
1125: return(Sminimal(pp));
1126: }
1127:
1.11 takayama 1128: def Sannfs2_laScala2(f) {
1129: local p,pp;
1130: p = Sannfs(f,"x,y");
1131: sm1(" p 0 get { [(x) (y) (Dx) (Dy)] laplace0 } map /p set ");
1132: p = [p];
1133: Sweyl("x,y",[["x",1,"y",1,"Dx",1,"Dy",1,"h",1],
1134: ["x",-1,"y",-1,"Dx",1,"Dy",1]]);
1135: pp = Map(p[0],"Spoly");
1136: return(Sminimal(pp));
1137: }
1138:
1.3 takayama 1139: def Sannfs3(f) {
1140: local p,pp;
1141: p = Sannfs(f,"x,y,z");
1.6 takayama 1142: sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");
1.3 takayama 1143: Sweyl("x,y,z",[["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]]);
1.6 takayama 1144: pp = Map(p,"Spoly");
1145: return(Sminimal_v(pp));
1.3 takayama 1146: }
1147:
1.10 takayama 1148: HelpAdd(["Sannfs3",
1149: ["Sannfs3(f) constructs the V-minimal free resolution for the weight (-1,1)",
1150: "of the Laplace transform of the annihilating ideal of the polynomial f in x,y,z.",
1151: "See also Sminimal_v, Sannfs2.",
1152: "Example: a=Sannfs3(\"x^3-y^2*z^2\");",
1153: " b=a[0]; sm1_pmat(b);",
1154: " b[1]*b[0]: b[2]*b[1]:"]]);
1155:
1.2 takayama 1156: /*
1157: The betti numbers of most examples are 2,1. (0-th and 1-th).
1158: a=Sannfs2("x*y*(x+y-1)"); ==> The betti numbers are 3, 2.
1159: a=Sannfs2("x^3-y^2-x"); : it causes an error. It should be fixed.
1.3 takayama 1160: a=Sannfs2("x*y*(x-y)"); : it causes an error. It should be fixed.
1.2 takayama 1161:
1162: */
1163:
1.11 takayama 1164: def Sannfs3_laScala2(f) {
1165: local p,pp;
1166: p = Sannfs(f,"x,y,z");
1167: sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");
1168: Sweyl("x,y,z",[["x",1,"y",1,"z",1,"Dx",1,"Dy",1,"Dz",1,"h",1],
1169: ["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]]);
1170: pp = Map(p,"Spoly");
1171: return(Sminimal(pp));
1172: }
1.5 takayama 1173:
1174:
1.6 takayama 1175: /* The below does not use LaScala-Stillman's algorithm. */
1.5 takayama 1176: def Sschreyer(g) {
1177: local rf, tower, reductionTable, skel, redundantTable, bases,
1178: strategy, maxOfStrategy, height, level, n, i,
1179: freeRes,place, f, reducer,pos, redundant_seq,bettiTable,freeResV,ww,
1180: redundantTable_ordinary, redundant_seq_ordinary,
1.6 takayama 1181: reductionTable_tmp,c2,ii,nn, m,ii, jj, reducerBase;
1.5 takayama 1182: /* extern WeightOfSweyl; */
1183: ww = WeightOfSweyl;
1184: Print("WeghtOfSweyl="); Println(WeightOfSweyl);
1185: rf = SresolutionFrameWithTower(g);
1186: redundant_seq = 1; redundant_seq_ordinary = 1;
1187: tower = rf[1];
1188: reductionTable = SgenerateTable(tower);
1189: skel = rf[2];
1190: redundantTable = SnewArrayOfFormat(rf[1]);
1191: redundantTable_ordinary = SnewArrayOfFormat(rf[1]);
1192: reducer = SnewArrayOfFormat(rf[1]);
1193: freeRes = SnewArrayOfFormat(rf[1]);
1194: bettiTable = SsetBettiTable(rf[1],g);
1195:
1196: height = Length(reductionTable);
1197: for (level = 0; level < height; level++) {
1198: n = Length(reductionTable[level]);
1199: for (i=0; i<n; i++) {
1200: Println([level,i]);
1201: Print("Processing "); Print([level,i]);
1202: if (level == 0) {
1203: if (IsNull(redundantTable[level,i])) {
1204: bases = freeRes[level];
1205: /* Println(["At floor : GB=",i,bases,tower[0,i]]); */
1206: pos = SwhereInGB(tower[0,i],rf[3,0]);
1207: bases[i] = rf[3,0,pos];
1208: /* redundantTable[level,i] = 0;
1209: redundantTable_ordinary[level,i] = 0; */
1210: freeRes[level] = bases;
1211: /* Println(["GB=",i,bases,tower[0,i]]); */
1212: }
1213: }else{ /* level >= 1 */
1214: if (IsNull(redundantTable[level,i])) {
1215: bases = freeRes[level];
1216: f = SpairAndReduction2(skel,level,i,freeRes,tower,
1217: ww,redundantTable);
1218: if (f[0] != Poly("0")) {
1219: place = f[3];
1220: /* (level-1, place) is the place for f[0],
1221: which is a newly obtained GB. */
1222: #ifdef ORDINARY
1223: redundantTable[level-1,place] = redundant_seq;
1224: redundant_seq++;
1225: #else
1226: if (f[4] > f[5]) {
1227: /* Zero in the gr-module */
1228: Print("v-degree of [org,remainder] = ");
1229: Println([f[4],f[5]]);
1230: Print("[level,i] = "); Println([level,i]);
1231: redundantTable[level-1,place] = 0;
1232: }else{
1233: redundantTable[level-1,place] = redundant_seq;
1234: redundant_seq++;
1235: }
1236: #endif
1237: redundantTable_ordinary[level-1,place]
1238: =redundant_seq_ordinary;
1239: redundant_seq_ordinary++;
1240: bases[i] = SunitOfFormat(place,f[1])-f[1]; /* syzygy */
1241: /* redundantTable[level,i] = 0;
1242: redundantTable_ordinary[level,i] = 0; */
1243: /* i must be equal to f[2], I think. Double check. */
1244:
1245: /* Correction Of Constant */
1.9 takayama 1246: /* Correction of syzygy */
1.7 takayama 1247: c2 = f[6]; /* or -f[6]? Double check. */
1248: Print("c2="); Println(c2);
1.5 takayama 1249: nn = Length(bases);
1250: for (ii=0; ii<nn;ii++) {
1.8 takayama 1251: if ((ii != i) && (! IsNull(bases[ii]))) {
1.7 takayama 1252: m = Length(bases[ii]);
1253: for (jj=0; jj<m; jj++) {
1254: if (jj != place) {
1255: bases[ii,jj] = bases[ii,jj]*c2;
1256: }
1257: }
1.5 takayama 1258: }
1259: }
1260:
1.7 takayama 1261: Print("Old freeRes[level] = "); sm1_pmat(freeRes[level]);
1.5 takayama 1262: freeRes[level] = bases;
1.7 takayama 1263: Print("New freeRes[level] = "); sm1_pmat(freeRes[level]);
1.6 takayama 1264:
1265: /* Update the freeRes[level-1] */
1.7 takayama 1266: Print("Old freeRes[level-1] = "); sm1_pmat(freeRes[level-1]);
1.6 takayama 1267: bases = freeRes[level-1];
1268: bases[place] = f[0];
1269: freeRes[level-1] = bases;
1.7 takayama 1270: Print("New freeRes[level-1] = "); sm1_pmat(freeRes[level-1]);
1.6 takayama 1271:
1.9 takayama 1272: reducer[level-1,place] = f[1]-SunitOfFormat(place,f[1]);
1273: /* This reducer is different from that of SlaScala(). */
1274:
1275: reducerBasis = reducer[level-1];
1276: nn = Length(reducerBasis);
1277: for (ii=0; ii<nn;ii++) {
1278: if ((ii != place) && (! IsNull(reducerBasis[ii]))) {
1279: m = Length(reducerBasis[ii]);
1280: for (jj=0; jj<m; jj++) {
1281: if (jj != place) {
1282: reducerBasis[ii,jj] = reducerBasis[ii,jj]*c2;
1283: }
1284: }
1285: }
1286: }
1287: reducer[level-1] = reducerBasis;
1288:
1.5 takayama 1289: }else{
1290: /* redundantTable[level,i] = 0; */
1291: bases = freeRes[level];
1292: bases[i] = f[1]; /* Put the syzygy. */
1293: freeRes[level] = bases;
1294: }
1295: } /* end of level >= 1 */
1296: }
1297: } /* i loop */
1.6 takayama 1298:
1299: /* Triangulate reducer */
1300: if (level >= 1) {
1301: Println(" ");
1302: Print("Triangulating reducer at level "); Println(level-1);
1.9 takayama 1303: Println("freeRes[level]="); sm1_pmat(freeRes[level]);
1.6 takayama 1304: reducerBase = reducer[level-1];
1305: Print("reducerBase="); Println(reducerBase);
1.9 takayama 1306: Println("Compare freeRes[level] and reducerBase (put -1)");
1.6 takayama 1307: m = Length(reducerBase);
1308: for (ii=m-1; ii>=0; ii--) {
1309: if (!IsNull(reducerBase[ii])) {
1310: for (jj=ii-1; jj>=0; jj--) {
1311: if (!IsNull(reducerBase[jj])) {
1312: if (!IsZero(reducerBase[jj,ii])) {
1.9 takayama 1313: /* reducerBase[ii,ii] should be always constant. */
1314: reducerBase[jj] = reducerBase[ii,ii]*reducerBase[jj]-reducerBase[jj,ii]*reducerBase[ii];
1.6 takayama 1315: }
1316: }
1317: }
1318: }
1319: }
1320: Println("New reducer");
1321: sm1_pmat(reducerBase);
1322: reducer[level-1] = reducerBase;
1323: }
1324:
1.5 takayama 1325: } /* level loop */
1326: n = Length(freeRes);
1327: freeResV = SnewArrayOfFormat(freeRes);
1328: for (i=0; i<n; i++) {
1329: bases = freeRes[i];
1330: bases = Sbases_to_vec(bases,bettiTable[i]);
1331: freeResV[i] = bases;
1332: }
1.6 takayama 1333:
1334: /* Mark the non-redundant elements. */
1335: for (i=0; i<n; i++) {
1336: m = Length(redundantTable[i]);
1337: for (jj=0; jj<m; jj++) {
1338: if (IsNull(redundantTable[i,jj])) {
1339: redundantTable[i,jj] = 0;
1340: }
1341: }
1342: }
1343:
1344:
1.5 takayama 1345: return([freeResV, redundantTable,reducer,bettiTable,redundantTable_ordinary]);
1346: }
1347:
1348: def SpairAndReduction2(skel,level,ii,freeRes,tower,ww,redundantTable) {
1349: local i, j, myindex, p, bases, tower2, gi, gj,
1350: si, sj, tmp, t_syz, pos, ans, ssp, syzHead,pos2,
1351: vdeg,vdeg_reduced,n,c2;
1.6 takayama 1352: Println("SpairAndReduction2 : -------------------------");
1.5 takayama 1353:
1354: if (level < 1) Error("level should be >= 1 in SpairAndReduction.");
1355: p = skel[level,ii];
1356: myindex = p[0];
1357: i = myindex[0]; j = myindex[1];
1358: bases = freeRes[level-1];
1359: Println(["p and bases ",p,bases]);
1360: if (IsNull(bases[i]) || IsNull(bases[j])) {
1361: Println([level,i,j,bases[i],bases[j]]);
1362: Error("level, i, j : bases[i], bases[j] must not be NULL.");
1363: }
1364:
1365: tower2 = StowerOf(tower,level-1);
1366: SsetTower(tower2);
1.14 takayama 1367: Println(["level=",level]);
1368: Println(["tower2=",tower2]);
1.5 takayama 1369: /** sm1(" show_ring "); */
1370:
1371: gi = Stoes_vec(bases[i]);
1372: gj = Stoes_vec(bases[j]);
1373:
1374: ssp = Sspolynomial(gi,gj);
1375: si = ssp[0,0];
1376: sj = ssp[0,1];
1377: syzHead = si*es^i;
1378: /* This will be the head term, I think. But, double check. */
1379: Println([si*es^i,sj*es^j]);
1380:
1381: Print("[gi, gj] = "); Println([gi,gj]);
1382: sm1(" [(Homogenize)] system_variable message ");
1383: Print("Reduce the element "); Println(si*gi+sj*gj);
1384: Print("by "); Println(bases);
1385:
1386: tmp = Sreduction(si*gi+sj*gj, bases);
1387:
1388: Print("result is "); Println(tmp);
1.6 takayama 1389: if (!IsZero(tmp[0])) {
1390: Print("Error: base = ");
1391: Println(Map(bases,"Stoes_vec"));
1392: Error("SpairAndReduction2: the remainder should be zero. See tmp. tower2. show_ring.");
1393: }
1.5 takayama 1394: t_syz = tmp[2];
1395: si = si*tmp[1]+t_syz[i];
1396: sj = sj*tmp[1]+t_syz[j];
1397: t_syz[i] = si;
1398: t_syz[j] = sj;
1399:
1400: c2 = null;
1401: /* tmp[0] must be zero */
1402: n = Length(t_syz);
1403: for (i=0; i<n; i++) {
1.6 takayama 1404: if (IsConstant(t_syz[i])){
1405: if (!IsZero(t_syz[i])) {
1.5 takayama 1406: if (IsNull(redundantTable[level-1,i])) {
1407: /* i must equal to pos2 below. */
1408: c2 = -t_syz[i];
1.6 takayama 1409: tmp[0] = c2*Stoes_vec(freeRes[level-1,i]);
1.5 takayama 1410: t_syz[i] = 0;
1.6 takayama 1411: /* tmp[0] = t_syz . g */
1.5 takayama 1412: /* break; does not work. Use */
1413: i = n;
1414: }
1.6 takayama 1415: }
1.5 takayama 1416: }
1417: }
1418:
1419: /* This is essential part for V-minimal resolution. */
1420: /* vdeg = SvDegree(si*gi+sj*gj,tower,level-1,ww); */
1421: vdeg = SvDegree(si*gi,tower,level-1,ww);
1422: vdeg_reduced = SvDegree(tmp[0],tower,level-1,ww);
1423: Print("vdegree of the original = "); Println(vdeg);
1424: Print("vdegree of the remainder = "); Println(vdeg_reduced);
1425:
1.11 takayama 1426: if (!IsNull(vdeg_reduced)) {
1427: if (vdeg_reduced < vdeg) {
1428: Println("--- Special in V-minimal!");
1429: Println(tmp[0]);
1430: Println("syzygy="); sm1_pmat(t_syz);
1431: Print("[vdeg, vdeg_reduced] = "); Println([vdeg,vdeg_reduced]);
1432: }
1433: }
1434:
1.14 takayama 1435: SsetTower(StowerOf(tower,level));
1436: pos = SwhereInTower(syzHead,tower[level]);
1.11 takayama 1437:
1.14 takayama 1438: SsetTower(StowerOf(tower,level-1));
1.5 takayama 1439: pos2 = SwhereInTower(tmp[0],tower[level-1]);
1440: ans = [tmp[0],t_syz,pos,pos2,vdeg,vdeg_reduced,c2];
1441: /* pos is the place to put syzygy at level. */
1442: /* pos2 is the place to put a new GB at level-1. */
1443: Println(ans);
1.14 takayama 1444: Println("--- end of SpairAndReduction2 ");
1.5 takayama 1445: return(ans);
1446: }
1.6 takayama 1447:
1.10 takayama 1448: HelpAdd(["Sminimal_v",
1449: ["It constructs the V-minimal free resolution from the Schreyer resolution",
1450: "step by step.",
1.14 takayama 1451: "This code still contains bugs. It sometimes outputs wrong answer.",
1.10 takayama 1452: "Example: Sweyl(\"x,y\",[[\"x\",-1,\"y\",-1,\"Dx\",1,\"Dy\",1]]);",
1453: " v=[[2*x*Dx + 3*y*Dy+6, 0],",
1454: " [3*x^2*Dy + 2*y*Dx, 0],",
1455: " [0, x^2+y^2],",
1456: " [0, x*y]];",
1457: " a=Sminimal_v(v);",
1458: " sm1_pmat(a[0]); b=a[0]; b[1]*b[0]:",
1459: "Note: a[0] is the V-minimal resolution. a[3] is the Schreyer resolution."]]);
1460:
1.14 takayama 1461: /* This code still contains bugs. It sometimes outputs wrong answer. */
1462: /* See test12() in minimal-test.k. */
1463: /* There may be remaining 1, too */
1.6 takayama 1464: def Sminimal_v(g) {
1465: local r, freeRes, redundantTable, reducer, maxLevel,
1466: minRes, seq, maxSeq, level, betti, q, bases, dr,
1.10 takayama 1467: betti_levelplus, newbases, i, j,qq,tminRes;
1.6 takayama 1468: r = Sschreyer(g);
1469: sm1_pmat(r);
1470: Debug_Sminimal_v = r;
1471: Println(" Return value of Schreyer(g) is set to Debug_Sminimal_v");
1472: /* Should I turn off the tower?? */
1473: freeRes = r[0];
1474: redundantTable = r[1];
1475: reducer = r[2];
1476: minRes = SnewArrayOfFormat(freeRes);
1477: seq = 0;
1478: maxSeq = SgetMaxSeq(redundantTable);
1479: maxLevel = Length(freeRes);
1480: for (level = 0; level < maxLevel; level++) {
1481: minRes[level] = freeRes[level];
1482: }
1483: for (level = 0; level < maxLevel; level++) {
1484: betti = Length(freeRes[level]);
1485: for (q = betti-1; q>=0; q--) {
1486: if (redundantTable[level,q] > 0) {
1487: Print("[seq,level,q]="); Println([seq,level,q]);
1488: if (level < maxLevel-1) {
1489: bases = freeRes[level+1];
1490: dr = reducer[level,q];
1.9 takayama 1491: /* dr[q] = -1; We do not need this in our reducer format. */
1492: /* dr[q] should be a non-zero constant. */
1.6 takayama 1493: newbases = SnewArrayOfFormat(bases);
1494: betti_levelplus = Length(bases);
1495: /*
1496: bases[i,j] ---> bases[i,j]+bases[i,q]*dr[j]
1497: */
1498: for (i=0; i<betti_levelplus; i++) {
1.9 takayama 1499: newbases[i] = dr[q]*bases[i] - bases[i,q]*dr;
1.6 takayama 1500: }
1501: Println(["level, q =", level,q]);
1502: Println("bases="); sm1_pmat(bases);
1503: Println("dr="); sm1_pmat(dr);
1504: Println("newbases="); sm1_pmat(newbases);
1505: minRes[level+1] = newbases;
1506: freeRes = minRes;
1507: #ifdef DEBUG
1.9 takayama 1508: for (qq=q; qq<betti; qq++) {
1.6 takayama 1509: for (i=0; i<betti_levelplus; i++) {
1.9 takayama 1510: if ((!IsZero(newbases[i,qq])) && (redundantTable[level,qq] >0)) {
1.6 takayama 1511: Println(["[i,qq]=",[i,qq]," is not zero in newbases."]);
1512: Print("redundantTable ="); sm1_pmat(redundantTable[level]);
1513: Error("Stop in Sminimal for debugging.");
1514: }
1515: }
1516: }
1517: #endif
1518: }
1519: }
1520: }
1521: }
1.10 takayama 1522: tminRes = Stetris(minRes,redundantTable);
1523: return([SpruneZeroRow(tminRes), tminRes,
1.6 takayama 1524: [ minRes, redundantTable, reducer,r[3],r[4]],r[0]]);
1525: /* r[4] is the redundantTable_ordinary */
1526: /* r[0] is the freeResolution */
1527: }
1528:
1529: /* Sannfs2("x*y*(x-y)*(x+y)"); is a test problem */
1.10 takayama 1530: /* x y (x+y-1)(x-2), x^3-y^2, x^3 - y^2 z^2,
1531: x y z (x+y+z-1) seems to be interesting, because the first syzygy
1532: contains 1.
1533: */
1534:
1535: def CopyArray(m) {
1536: local ans,i,n;
1537: if (IsArray(m)) {
1538: n = Length(m);
1539: ans = NewArray(n);
1540: for (i=0; i<n; i++) {
1541: ans[i] = CopyArray(m[i]);
1542: }
1543: return(ans);
1544: }else{
1545: return(m);
1546: }
1547: }
1548: HelpAdd(["CopyArray",
1549: ["It duplicates the argument array recursively.",
1550: "Example: m=[1,[2,3]];",
1551: " a=CopyArray(m); a[1] = \"Hello\";",
1552: " Println(m); Println(a);"]]);
1553:
1554: def IsZeroVector(m) {
1555: local n,i;
1556: n = Length(m);
1557: for (i=0; i<n; i++) {
1558: if (!IsZero(m[i])) {
1559: return(false);
1560: }
1561: }
1562: return(true);
1563: }
1564:
1565: def SpruneZeroRow(res) {
1566: local minRes, n,i,j,m, base,base2,newbase,newbase2, newMinRes;
1567:
1568: minRes = CopyArray(res);
1569: n = Length(minRes);
1570: for (i=0; i<n; i++) {
1571: base = minRes[i];
1572: m = Length(base);
1573: if (i != n-1) {
1574: base2 = minRes[i+1];
1575: base2 = Transpose(base2);
1576: }
1577: newbase = [ ];
1578: newbase2 = [ ];
1579: for (j=0; j<m; j++) {
1580: if (!IsZeroVector(base[j])) {
1581: newbase = Append(newbase,base[j]);
1582: if (i != n-1) {
1583: newbase2 = Append(newbase2,base2[j]);
1584: }
1585: }
1586: }
1587: minRes[i] = newbase;
1588: if (i != n-1) {
1589: if (newbase2 == [ ]) {
1590: minRes[i+1] = [ ];
1591: }else{
1592: minRes[i+1] = Transpose(newbase2);
1593: }
1594: }
1595: }
1596:
1597: newMinRes = [ ];
1598: n = Length(minRes);
1599: i = 0;
1600: while (i < n ) {
1601: base = minRes[i];
1602: if (base == [ ]) {
1603: i = n; /* break; */
1604: }else{
1605: newMinRes = Append(newMinRes,base);
1606: }
1607: i++;
1608: }
1609: return(newMinRes);
1610: }
1611:
1612: def testAnnfs2(f) {
1613: local a,i,n;
1614: a = Sannfs2(f);
1615: b=a[0];
1616: n = Length(b);
1617: Println("------ V-minimal free resolution -----");
1618: sm1_pmat(b);
1619: Println("----- Is it complex? ---------------");
1620: for (i=0; i<n-1; i++) {
1621: Println(b[i+1]*b[i]);
1622: }
1623: return(a);
1624: }
1625: def testAnnfs3(f) {
1626: local a,i,n;
1627: a = Sannfs3(f);
1628: b=a[0];
1629: n = Length(b);
1630: Println("------ V-minimal free resolution -----");
1631: sm1_pmat(b);
1632: Println("----- Is it complex? ---------------");
1633: for (i=0; i<n-1; i++) {
1634: Println(b[i+1]*b[i]);
1635: }
1.11 takayama 1636: return(a);
1637: }
1638:
1639: def ToString_array(p) {
1640: local ans;
1641: if (IsArray(p)) {
1642: ans = Map(p,"ToString_array");
1643: }else{
1644: ans = ToString(p);
1645: }
1646: return(ans);
1647: }
1648:
1649: /* sm1_res_div([[x],[y]],[[x^2],[x*y],[y^2]],[x,y]): */
1650:
1651: def sm1_res_div(I,J,V) {
1652: I = ToString_array(I);
1653: J = ToString_array(J);
1654: V = ToString_array(V);
1655: sm1(" [[ I J] V ] res*div /FunctionValue set ");
1656: }
1657:
1658: /* It has not yet been working */
1659: def sm1_res_kernel_image(m,n,v) {
1660: m = ToString_array(m);
1661: n = ToString_array(n);
1662: v = ToString_array(v);
1663: sm1(" [m n v] res-kernel-image /FunctionValue set ");
1664: }
1665: def Skernel(m,v) {
1666: m = ToString_array(m);
1667: v = ToString_array(v);
1668: sm1(" [ m v ] syz /FunctionValue set ");
1669: }
1670:
1671: def test3() {
1672: local a1,a2,b1,b2;
1673: a1 = Sannfs3("x^3-y^2*z^2");
1674: a1 = a1[0];
1675: a2 = Sannfs3_laScala2("x^3-y^2*z^2");
1676: a2 = a2[0];
1677: b1 = a1[1];
1678: b2 = a2[1];
1679: sm1_pmat(b2);
1680: Println(" OVER ");
1681: sm1_pmat(b1);
1682: return([sm1_res_div(b2,b1,["x","y","z"]),b2,b1,a2,a1]);
1683: }
1684:
1685: def test4() {
1686: local a,b;
1687: a = Sannfs3_laScala2("x^3-y^2*z^2");
1688: b = a[0];
1689: sm1_pmat( sm1_res_kernel_image(b[0],b[1],[x,y,z]));
1690: sm1_pmat( sm1_res_kernel_image(b[1],b[2],[x,y,z]));
1691: return(a);
1692: }
1693:
1694: def sm1_gb(f,v) {
1695: f =ToString_array(f);
1696: v = ToString_array(v);
1697: sm1(" [f v] gb /FunctionValue set ");
1.13 takayama 1698: }
1699:
1.11 takayama 1700:
1.12 takayama 1701: def SisComplex(a) {
1702: local n,i,j,k,b,p,q;
1703: n = Length(a);
1704: for (i=0; i<n-1; i++) {
1705: if (Length(a[i+1]) != 0) {
1706: b = a[i+1]*a[i];
1707: p = Length(b); q = Length(b[0]);
1708: for (j=0; j<p; j++) {
1709: for (k=0; k<q; k++) {
1710: if (!IsZero(b[j,k])) {
1711: Print("Is is not complex at ");
1712: Println([i,j,k]);
1713: return(false);
1714: }
1715: }
1716: }
1717: }
1718: }
1719: return(true);
1.14 takayama 1720: }
1721:
1722: def IsExact_h(c,v) {
1723: local a;
1724: v = ToString_array(v);
1725: a = [c,v];
1726: sm1(a," isExact_h /FunctionValue set ");
1727: }
1728: HelpAdd(["IsExact_h",
1729: ["IsExact_h(complex,var): bool",
1730: "It checks the given complex is exact or not in D<h> (homogenized Weyl algebra)",
1731: "cf. ReParse"
1732: ]]);
1733:
1734: def ReParse(a) {
1735: local c;
1736: if (IsArray(a)) {
1737: c = Map(a,"ReParse");
1738: }else{
1739: sm1(a," toString . /c set");
1740: }
1741: return(c);
1742: }
1743: HelpAdd(["ReParse",
1744: ["Reparse(obj): obj",
1745: "It parses the given object in the current ring.",
1746: "Outputs from SlaScala, Sschreyer may cause a trouble in other functions,",
1747: "because it uses the Schreyer order.",
1748: "In this case, ReParse the outputs from these functions.",
1749: "cf. IsExaxt_h"
1750: ]]);
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