Annotation of OpenXM/src/k097/lib/restriction/deRham.k, Revision 1.2
1.2 ! takayama 1: /* $OpenXM: OpenXM/src/k097/lib/restriction/deRham.k,v 1.1 2006/10/01 05:49:22 takayama Exp $ */
1.1 takayama 2:
3: /*
4: Require: restriction.k, ox.k
5:
6: generic_bfct(ii,vv,dd,ww);
7: ex.
8: RingD("x,y");
9: generic_bfct([Dx^2+Dy^2-1,Dx*Dy-4],[x,y],[Dx,Dy],[1,1]):/
10: DeRham2(f);
11: DeRham3(f);
12: */
13:
14: def demoSendAsirCommand(a) {
15: a.executeString("load(\"bfct\");");
16: a.executeString(" def myann(F) { B=ann(eval_str(F)); print(B); return(map(dp_ptod,B,[hoge,x,y,z,s,hh,ee,dx,dy,dz,ds,dhh])); }; ");
17: a.executeString(" def myann0(F) { B=ann0(eval_str(F)); print(B); return(map(dp_ptod,B[1],[hoge,x,y,z,s,hh,ee,dx,dy,dz,ds,dhh])); }; ");
18: a.executeString(" def mybfct(F) { return(rtostr(bfct(eval_str(F)))); }; ");
19: a.executeString(" def mygeneric_bfct(F,VV,DD,WW) { print([F,VV,DD,WW]); return(generic_bfct(F,VV,DD,WW));}; ");
20: }
21:
22: if (Boundp("NoX")) {
23: as = Asir.generate(false);
24: }else{
25: as = Asir.generate();
26: }
27:
28: def demoReduction(v) {
29: if (v == true) {
30: sm1(" [(Verbose) 1] system_variable [(DebugReductionRed) 1] system_variable Onverbose ");
31: }else{
32: sm1(" [(Verbose) 0] system_variable [(DebugReductionRed) 0] system_variable Offverbose ");
33: }
34: }
35:
36: asssssir = as;
37: demoSendAsirCommand(as);
38: RingD("x,y,z,s");
39:
40: def asirBfunction(a,f) {
41: local p,b;
42: p = ToString(f);
43: Println(p);
44: b = a.rpc("mybfct",[p]);
45: sm1(" b . /b set ");
46: return(b);
47: }
48:
49: def asirAnnfsXYZ(a,f) {
50: local p,b;
51: RingD("x,y,z,s"); /* Fix!! See the definition of myann() */
52: p = ToString(f);
53: b = a.rpc("myann",[p]);
54: return(b);
55: }
56:
57:
58: def asir_generic_bfct(a,ii,vv,dd,ww) {
59: local ans;
60: ans = a.rpc_str("mygeneric_bfct",[ii,vv,dd,ww]);
61: return(ans);
62: }
63: /* a=startAsir();
64: asir_generic_bfct(a,[Dx^2+Dy^2-1,Dx*Dy-4],[x,y],[Dx,Dy],[1,1]): */
65:
66: def generic_bfct(ii,vv,dd,ww) {
67: local ans;
68: ans =asir_generic_bfct(asssssir,ii,vv,dd,ww);
69: return(ans);
70: }
71:
72: /* usage: misc/tmp/complex-ja.texi */
73: def ChangeRing(f) {
74: local r;
75: r = GetRing(f);
76: if (Tag(r) == 14) {
77: SetRing(r);
78: return(true);
79: }else{
80: return(false);
81: }
82: }
83:
84: def asir_BfRoots2(G) {
85: local bb,ans,ss;
86: sm1(" G flatten {dehomogenize} map /G set ");
87: ChangeRing(G);
88: ss = asir_generic_bfct(asssssir,G,[x,y],[Dx,Dy],[1,1]);
89: bb = [ss];
90: sm1(" bb 0 get findIntegralRoots { (universalNumber) dc } map /ans set ");
91: return([ans, bb]);
92: }
93: def asir_BfRoots3(G) {
94: local bb,ans,ss;
95: sm1(" G flatten {dehomogenize} map /G set ");
96: ChangeRing(G);
97: ss = asir_generic_bfct(asssssir,G,[x,y,z],[Dx,Dy,Dz],[1,1,1]);
98: bb = [ss];
99: sm1(" bb 0 get findIntegralRoots { (universalNumber) dc } map /ans set ");
100: return([ans, bb]);
101: }
102:
103: def asir_BfRoots4(G) {
104: local bb,ans,ss;
105: sm1(" G flatten {dehomogenize} map /G set ");
106: ChangeRing(G);
107: ss = asir_generic_bfct(asssssir,G,[x,y,z,vv],[Dx,Dy,Dz,Dvv],[0,0,0,1]);
108: bb = [ss];
109: sm1(" bb 0 get findIntegralRoots { (universalNumber) dc } map /ans set ");
110: return([ans, bb]);
111: }
112:
113: def findMinSol(f) {
114: sm1(" f (string) dc findIntegralRoots 0 get (universalNumber) dc /FunctionValue set ");
115: }
116:
117: def asirAnnXYZ(a,f) {
118: local p,b,b0,k1;
119: RingD("x,y,z,s"); /* Fix!! See the definition of myann() */
120: p = ToString(f);
121: b = a.rpc("myann",[p]);
122: Print("Annhilating ideal with s is "); Println(b);
123: b0 = asirBfunction(a,f);
124: Print("bfunction is "); Println(b0);
125: k1 = findMinSol(b0);
126: Print("Minimal integral root is "); Println(k1);
127: sm1(" b { [[(s). k1 (string) dc .]] replace } map /b set ");
128: return(b);
129: }
130:
131:
132: def nonquasi2(p,q) {
133: local s,ans,f;
134:
135: sm1("0 set_timer "); sm1(" oxNoX ");
136: asssssir.OnTimer();
137:
138: f = x^p+y^q+x*y^(q-1);
139: Print("f=");Println(f);
140: s = ToString(f);
141: sm1(" Onverbose ");
142: ans = asirAnnfsXYZ(asssssir,f);
143: sm1(" ans 0 get (ring) dc ring_def ");
144: sm1("[ ans { [[(s). (-1).]] replace } map ] /II set ");
145: Println("Step 1: Annhilating ideal (II)"); Println(II);
146: sm1(" II 0 get { [(x) (y) (Dx) (Dy) ] laplace0 } map /II set ");
147: Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
148: pp = Map(II,"Spoly");
149: Res = Sminimal(pp);
150: Res0 = Res[0];
151: Println("Step2: (-1,1)-minimal resolution (Res0) "); sm1_pmat(Res0);
152: /* R = BfRoots1(Res0[0],"x,y"); */
153: R = asir_BfRoots2(Res0[0]);
154: Println("Step3: computing the cohomology of the truncated complex.");
155: Print("Roots and b-function are "); Println(R);
156: R0 = R[0];
157: Ans=Srestall(Res0, ["x", "y"], ["x", "y"], R0[Length(R0)-1]);
158:
159: Println("Timing data: sm1 "); sm1(" 1 set_timer ");
160: Print(" ox_asir [CPU,GC]: ");Println(asssssir.OffTimer());
161:
162: Print("Answer is "); Println(Ans[0]);
163: return(Ans);
164: }
165:
166: def asirAnn0XYZ(a,f) {
167: local p,b,b0;
168: RingD("x,y,z,s"); /* Fix!! See the definition of myann() */
169: p = ToString(f);
170: b = a.rpc("myann0",[p]);
171: Print("Annhilating ideal of f^r is "); Println(b);
172: return(b);
173: }
174:
175:
176: def DeRham2(f) {
177: local s;
178:
179: sm1("0 set_timer "); sm1(" oxNoX ");
180: asssssir.OnTimer();
181:
182: s = ToString(f);
183: II = asirAnn0XYZ(asssssir,f);
184: Print("Step 1: Annhilating ideal (II)"); Println(II);
185: sm1(" II { [(x) (y) (Dx) (Dy) ] laplace0 } map /II set ");
186: Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
187: pp = Map(II,"Spoly");
188: Res = Sminimal(pp);
189: Res0 = Res[0];
190: Print("Step2: (-1,1)-minimal resolution (Res0) "); sm1_pmat(Res0);
191: /* R = BfRoots1(Res0[0],"x,y"); */
192: R = asir_BfRoots2(Res0[0]);
193: Println("Step3: computing the cohomology of the truncated complex.");
194: Print("Roots and b-function are "); Println(R);
195: R0 = R[0];
196: Ans=Srestall(Res0, ["x", "y"], ["x", "y"],R0[Length(R0)-1] );
197:
198: Println("Timing data: sm1 "); sm1(" 1 set_timer ");
199: Print(" ox_asir [CPU,GC]: ");Println(asssssir.OffTimer());
200:
201: Print("Answer is ");Println(Ans[0]);
202: return(Ans);
203: }
204: def DeRham3(f) {
205: local s;
206:
207: sm1("0 set_timer "); sm1(" oxNoX ");
208: asssssir.OnTimer();
209:
210: s = ToString(f);
211: II = asirAnn0XYZ(asssssir,f);
212: Print("Step 1: Annhilating ideal (II)"); Println(II);
213: sm1(" II { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /II set ");
214: Sweyl("x,y,z",[["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]]);
215: pp = Map(II,"Spoly");
216: Res = Sminimal(pp);
217: Res0 = Res[0];
218: Print("Step2: (-1,1)-minimal resolution (Res0) "); sm1_pmat(Res0);
219: /* R = BfRoots1(Res0[0],"x,y,z"); */
220: R = asir_BfRoots3(Res0[0]);
221: Println("Step3: computing the cohomology of the truncated complex.");
222: Print("Roots and b-function are "); Println(R);
223: R0 = R[0];
224: Ans=Srestall(Res0, ["x", "y", "z"], ["x", "y", "z"],R0[Length(R0)-1] );
225:
226: Println("Timing data: sm1 "); sm1(" 1 set_timer ");
227: Print(" ox_asir [CPU,GC]: ");Println(asssssir.OffTimer());
228:
229: Print("Answer is ");Println(Ans[0]);
230: return(Ans);
231: }
232:
233:
234: /* test data
235:
236: NoX=true;
237: nonquasi2(4,5);
238: nonquasi2(4,6);
239: nonquasi2(4,7);
240: nonquasi2(4,8);
241: nonquasi2(4,9);
242: nonquasi2(4,10);
243:
244: nonquasi2(5,6);
245: nonquasi2(6,7);
246: nonquasi2(7,8);
247: nonquasi2(8,9);
248: nonquasi2(9,10);
249: */
250:
251: P2 = [
252: "x^3-y^2",
253: "y^2-x^3-x-1",
254: "y^2-x^5-x-1",
255: "y^2-x^7-x-1",
256: "y^2-x^9-x-1",
257: "y^2-x^11-x-1"
258: ];
259:
260: P3 = [
261: "x^3-y^2*z^2",
262: "x^2*z+y^3+y^2*z+z^3",
263: "y*z^2+x^3+x^2*y^2+y^6",
264: "x*z^2+x^2*y+x*y^3+y^5"
265: ];
266:
267:
268: def diff_tmp(ff,xx) {
269: local g;
270: g = xx*ff;
271: return( Replace(g,[[xx,Poly("0")],[h,Poly("1")]]));
272: }
273:
274: def Localize3WithAsir(I,f) {
275: local s;
276:
277: sm1("0 set_timer "); sm1(" oxNoX ");
278: asssssir.OnTimer();
279:
280: RingD("x,y,z,vv");
281: /* BUG: use of RingD("x,y,z,v") causes an expected error.
282: [x2,x3,x4,x4] in mygeneric_bfct. (should be [x2,x3,x4,x5]).
283: */
284: f = ReParse(f);
285: I = ReParse(I);
286: /* Test data. */
287: /*
288: f = x^3-y^2*z^2;
289: I = [f^2*Dx-3*x^2, f^2*Dy-2*y*z^2, f^2*Dz-2*y^2*z];
290:
291: f = x^3-y^2;
292: I = [f^2*Dx-diff_tmp(f,Dx), f^2*Dy-diff_tmp(f,Dy), Dz];
293: */
294:
295: r1 = Dx-vv^2*diff_tmp(f,Dx)*Dvv;
296: r2 = Dy-vv^2*diff_tmp(f,Dy)*Dvv;
297: r3 = Dz-vv^2*diff_tmp(f,Dz)*Dvv;
298: II = ReParse(I);
299: sm1(" II { [[(Dx). r1] [(Dy). r2] [(Dz). r3]] replace } map dehomogenize /II set ");
300:
301: Print("Step 1: phi(J)"); Println(II);
302: II = Join(II,[vv*f-1]);
303: Print("Step 2: <phi(J),vf-1>"); Println(II);
304:
305: Println("Step3: computing the integral.");
306: sm1(" II { [(x) (y) (z) (vv) (Dx) (Dy) (Dz) (Dvv)] laplace0 } map /JJ set ");
307: Sweyl("x,y,z,vv",[["vv",-1,"Dvv",1]]);
308: pp = Map(JJ,"Spoly");
309: R = asir_BfRoots4(pp);
310: Print("Roots and b-function are "); Println(R);
311:
312: R0 = R[0];
313: k1 = R0[Length(R0)-1];
314: sm1(" [(parse) (intw.sm1) pushfile] extension /intw.verbose 1 def ");
315: sm1(" [II [(vv) (x) (y) (z)] [(vv) -1 (Dvv) 1] k1 (integer) dc] integral-k1 /Ans set ");
316:
317: Println("Timing data: sm1 "); sm1(" 1 set_timer ");
318: Print(" ox_asir [CPU,GC]: ");Println(asssssir.OffTimer());
319:
320: Print("Answer is ");Println(Ans);
321: return(Ans);
322: }
323:
324:
325: def Ltest2() {
326: RingD("x,y,z");
327: f = x^3-y^2;
328: I = [f^2*Dx-diff_tmp(f,Dx), f^2*Dy-diff_tmp(f,Dy), Dz];
329: return( Localize3WithAsir(I,f) );
330: }
1.2 ! takayama 331:
! 332: /* The polynomial variable name w cannot be used. Why? ww also causes a trouble, because
! 333: perhaps there is a local variable ww. ww --> www. 2014.11.27 */
! 334: def asir_BfRoots4www(G) {
! 335: local bb,ans,ss;
! 336: sm1(" G flatten {dehomogenize} map /G set ");
! 337: ChangeRing(G);
! 338: ss = asir_generic_bfct(asssssir,G,[x,y,z,www],[Dx,Dy,Dz,Dwww],[1,1,1,1]);
! 339: bb = [ss];
! 340: sm1(" bb 0 get findIntegralRoots { (universalNumber) dc } map /ans set ");
! 341: return([ans, bb]);
! 342: }
! 343: def asirAnn0XYZWWW(a,f) {
! 344: local p,b,b0;
! 345: RingD("x,y,z,www,s"); /* Fix!! See the definition of myann0www() */
! 346: p = ToString(f);
! 347: a.executeString(" def myann0www(F) { B=ann0(eval_str(F)); print(B); return(map(dp_ptod,B[1],[hoge,x,y,z,www,s,hh,ee,dx,dy,dz,dwww,ds,dhh])); }; ");
! 348: b = a.rpc("myann0www",[p]);
! 349: Print("Annhilating ideal of f^r is "); Println(b);
! 350: return(b);
! 351: }
! 352:
! 353: def DeRham4(f) {
! 354: local s;
! 355:
! 356: sm1("0 set_timer "); sm1(" oxNoX ");
! 357: asssssir.OnTimer();
! 358:
! 359: s = ToString(f);
! 360: II = asirAnn0XYZWWW(asssssir,f);
! 361: Print("Step 1: Annhilating ideal (II)"); Println(II);
! 362: sm1(" II { [(x) (y) (z) (www) (Dx) (Dy) (Dz) (Dwww)] laplace0 } map /II set ");
! 363: Sweyl("x,y,z,www",[["x",-1,"y",-1,"z",-1,"www",-1,"Dx",1,"Dy",1,"Dz",1,"Dwww",1]]);
! 364: pp = Map(II,"Spoly");
! 365: Res = Sminimal(pp);
! 366: Res0 = Res[0];
! 367: Print("Step2: (-1,1)-minimal resolution (Res0) "); sm1_pmat(Res0);
! 368:
! 369: R = asir_BfRoots4www(Res0[0]);
! 370: Println("Step3: computing the cohomology of the truncated complex.");
! 371: Print("Roots and b-function are "); Println(R);
! 372: R0 = R[0];
! 373: Ans=Srestall(Res0, ["x", "y", "z","www"], ["x", "y", "z","www"],R0[Length(R0)-1] );
! 374:
! 375: Println("Timing data: sm1 "); sm1(" 1 set_timer ");
! 376: Print(" ox_asir [CPU,GC]: ");Println(asssssir.OffTimer());
! 377:
! 378: Print("Answer is ");Println(Ans[0]);
! 379: return(Ans);
! 380: }
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