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1.1 ! noro 1: /* $OpenXM: OpenXM/src/asir99/lib/primdec,v 1.1.1.1 1999/11/10 08:12:30 noro Exp $ */
! 2: /* Primary decomposition & Radical decomposition program */
! 3: /* written by T.Shimoyama, Fujitsu Lab. Date: 1995.10.12 */
! 4:
! 5: /* comments about flags *
! 6: PRIMEORD term order of Groebner basis in primedec
! 7: PRIMAORD term order of Groebner basis in primadec
! 8: PRINTLOG print out log (1 (simple) -- 5 (pricise))
! 9: SHOWTIME if 1 : print out timings and data
! 10: ISOLATED if 1 : compute only isolated primary components
! 11: NOEMBEDDED if 1 : compute only pseudo-primary components
! 12: NOINITGB if 1 : no initial G-base computation
! 13: REDUNDANT if 1 : no redundancy check
! 14: COMPLETE if 1 : use complete criterion of redundancy
! 15: COMMONCHECK if 1 : redundancy check by intersection (in gr_fctr_sub)
! 16: SELECTFLAG selection strategy of separators (0 -- 3)
! 17: */
! 18:
! 19: if (vtype(minipoly) != 3) load("gr")$$
! 20:
! 21: #define GR(R,F,V,O) T2=newvect(4,time());R=dp_gr_main(F,V,0,0,O);GRTIME+=newvect(4,time())-T2;
! 22: #define HGRM(R,F,V,O) T2=newvect(4,time());R=dp_gr_main(F,V,1,1,O);GRTIME+=newvect(4,time())-T2;
! 23: #define NF(R,IN,F,G,O) T2=newvect(4,time());R=dp_nf(IN,F,G,O);NFTIME+=newvect(4,time())-T2;
! 24:
! 25: /* optional flags */
! 26: extern PRIMAORD,PRIMEORD,PRINTLOG,SHOWTIME,NOINITGB$
! 27: extern COMMONCHECK,ISOLATED,NOEMBEDDED,REDUNDANT,SELECTFLAG,COMPLETE$
! 28: extern EXTENDED,CONTINUE,BSAVE,MAXALGIND,NOSPECIALDEC$
! 29: def primflags() {
! 30: print("PRIMAORD,PRIMEORD,PRINTLOG,SHOWTIME,NOINITGB,ISOLATED,NOEMBEDDED,COMMONCHECK");
! 31: print("REDUNDANT,SELECTFLAG,COMPLETE,EXTENDED,CONTINUE,BSAVE,MAXALGIND,NOSPECIALDEC");}
! 32: PRIMAORD=0$ PRIMEORD=3$
! 33:
! 34: /* global variables */
! 35: extern DIRECTRY,COMMONIDEAL,NOMRADDEC,NOMISOLATE,RADDECTIME$
! 36: extern MISTIME,ISORADDEC,GRTIME,NFTIME$
! 37:
! 38:
! 39: /* primary ideal decomposition of ideal(F) */
! 40: /* return the list of [P,Q] where Q is P-primary component of ideal(F) */
! 41: def primadec(F,VL)
! 42: {
! 43: if ( !VL ) {
! 44: print("invalid variables"); return 0; }
! 45: if ( !F ) {
! 46: print("invalid argument"); return 0; }
! 47: if ( F == [] )
! 48: return [];
! 49: if ( length(F) == 1 && type(F[0]) == 1 )
! 50: return [[1],[1]];
! 51: NOMRADDEC = NOMISOLATE = 0; RADDECTIME = newvect(4); T0 = newvect(4,time());
! 52: GRTIME = newvect(4); NFTIME = newvect(4); MISTIME = newvect(4);
! 53:
! 54: R = primadec_main(F,[["begin",F,F,[1],[]]],[],VL);
! 55:
! 56: if ( PRINTLOG ) {
! 57: print(""); print("primary decomposition done.");
! 58: }
! 59: if ( PRINTLOG || SHOWTIME ) {
! 60: print("number of radical decompositions = ",0); print(NOMRADDEC);
! 61: print("number of primary components = ",0); print(length(R),0);
! 62: print(" ( isolated = ",0); print(NOMISOLATE,0); print(" )");
! 63: print("total time of m.i.s. computation = ",0);
! 64: print(MISTIME[0],0); GT = MISTIME[1];
! 65: if ( GT ) { print(" + gc ",0);print(GT); } else print("");
! 66: print("total time of n-form computation = ",0);
! 67: print(NFTIME[0],0); GT = NFTIME[1];
! 68: if ( GT ) { print(" + gc ",0);print(GT); } else print("");
! 69: print("total time of G-base computation = ",0);
! 70: print(GRTIME[0],0); GT = GRTIME[1];
! 71: if ( GT ) { print(" + gc ",0);print(GT); } else print("");
! 72: print("total time of radical decomposition = ",0);
! 73: print(RADDECTIME[0],0); GT = RADDECTIME[1];
! 74: if ( GT ) { print(" + gc ",0);print(GT,0); }
! 75: print(" (iso ",0); print(ISORADDEC[0],0); GT = ISORADDEC[1];
! 76: if ( GT ) { print(" +gc ",0);print(GT,0); } print(")");
! 77: print("total time of primary decomposition = ",0);
! 78: TT = newvect(4,time())-T0; print(TT[0],0);
! 79: if ( TT[1] ) { print(" + gc ",0);print(TT[1]); } else print("");
! 80: }
! 81: return R;
! 82: }
! 83:
! 84: /* prime ideal decomposition of radical(F) */
! 85: /* return the list of prime components of radical(F) */
! 86:
! 87: def primedec(F,VL)
! 88: {
! 89: if ( !VL ) {
! 90: print("invalid variables"); return 0; }
! 91: if ( !F ) {
! 92: print("invalid argument"); return 0; }
! 93: if ( F == [] )
! 94: return [];
! 95: GRTIME = newvect(4);
! 96: T0 = newvect(4,time());
! 97: if ( !NOINITGB ) {
! 98: if ( PRINTLOG ) {
! 99: print("G-base computation w.r.t ordering = ",0);
! 100: print(PRIMEORD);
! 101: }
! 102: T1 = newvect(4,time());
! 103: HGRM(F,F,VL,PRIMEORD);
! 104: Tg = newvect(4,time())-T1;
! 105: if ( PRINTLOG > 0 ) {
! 106: print(" G-base time = ",0); print(Tg[0],0);
! 107: if ( Tg[1] ) { print(" + gc : ",0); print(Tg[1]); }
! 108: else print("");
! 109: }
! 110: }
! 111:
! 112: R = primedec_main([F],VL);
! 113:
! 114: Ta = newvect(4,time())-T0;
! 115: if ( PRINTLOG || SHOWTIME ) {
! 116: print("number of prime components = ",0); print(length(R));
! 117: print("G-base time = ",0); print(GRTIME[0],0);
! 118: if ( GRTIME[1] ) { print(" + gc : ",0); print(GRTIME[1]); }
! 119: else print("");
! 120: print("total time = ",0); print(Ta[0],0);
! 121: if ( Ta[1] ) { print(" + gc : ",0); print(Ta[1]); } else print("");
! 122: }
! 123: return R;
! 124: }
! 125:
! 126: /* main procedure for primary decomposition. */
! 127: /* F : ideal, VL : variable list, REMS : remaining components */
! 128: def primadec_main(F,REMS,H,VL)
! 129: {
! 130: DEC = RES = [];
! 131: for (D = [], I = 0; I < length(REMS); I++) {
! 132: MARK = REMS[I][0]; WF = REMS[I][1]; WP = REMS[I][2]; SC = REMS[I][3];
! 133: if ( !NOINITGB || MARK != "begin" ) {
! 134: ORD = PRIMAORD;
! 135: if ( PRINTLOG > 1 ) {
! 136: if ( MARK != "begin" ) print("");
! 137: print("G-base computation w.r.t ordering = ",0);
! 138: print(ORD);
! 139: }
! 140: T1 = newvect(4,time());
! 141: /* G-base of ideal */
! 142: GR(GF,WF,VL,ORD);
! 143: if ( MARK != "begin" && ( COMPLETE || EXTENDED ) ) {
! 144: if ( SC!=[1] && SC!=[-1] ) {
! 145: LG = localize(WF,SC,VL,ORD); /* VR_s\cap R */
! 146: if ( EXTENDED ) { GF = LG; }
! 147: } else
! 148: LG = GF;
! 149: if ( idealinc(H,LG,VL,ORD) ) {
! 150: if ( PRINTLOG ) {
! 151: print("eliminate the remaining component ",0);
! 152: print("by complete criterion");
! 153: }
! 154: continue; /* complete criterion */
! 155: }
! 156: }
! 157: /* G-base of radical */
! 158: if ( MARK == "begin" ) {
! 159: RA = ["begin",GF];
! 160: } else if ( MARK == "ext" ) {
! 161: if ( WF==WP || idealinc(WP,GF,VL,ORD) )
! 162: RA = ["ext",GF];
! 163: else {
! 164: if ( EXTENDED ) {
! 165: GA = localize(WP,SC,VL,PRIMEORD);
! 166: } else {
! 167: GR(GA,WP,VL,PRIMEORD);
! 168: }
! 169: RA = ["ext",GA];
! 170: }
! 171: } else if ( MARK == "sep" ) {
! 172: for (U=[], T=WP,J=length(T)-1;J>=0;J--) {
! 173: if ( EXTENDED ) {
! 174: GA = localize(T[J],SC,VL,PRIMEORD);
! 175: } else {
! 176: GR(GA,T[J],VL,PRIMEORD);
! 177: }
! 178: if (GA != [1] && GA != [-1])
! 179: U = cons(GA,U);
! 180: }
! 181: RA = ["sep",U];
! 182: } else debug;
! 183: Tg = newvect(4,time())-T1;
! 184: if ( PRINTLOG > 1 ) {
! 185: print(" G-base time = ",0); print(Tg[0],0);
! 186: if ( Tg[1] ) { print(" + gc : ",0); print(Tg[1]); }
! 187: else print("");
! 188: }
! 189: } else {
! 190: GF = F; /* NOINITGB = 1 */
! 191: RA = ["begin",GF];
! 192: }
! 193: if ( PRINTLOG ) {
! 194: if ( MARK == "begin" ) {
! 195: print("primary decomposition of the ideal");
! 196: } else { /* at the begining */
! 197: print("");
! 198: print("primary decomposition of the remaining component");
! 199: }
! 200: if ( MARK == "begin" && PRINTLOG > 1 ) { /* at the begining */
! 201: print(" ideal = ",0);
! 202: print(WF);
! 203: } else if ( PRINTLOG >= 4 ) {
! 204: print(" remaining component = ",0);
! 205: print(GF);
! 206: }
! 207: }
! 208:
! 209: /* input: init, generator, G-base, radical, intersection, sep.cond.*/
! 210: /* output: primary comp. remaining comp. */
! 211: Y = isolated(F,WF,GF,RA,REMS[I][4],SC,VL);
! 212:
! 213: D = append(D,Y[0]);
! 214: if ( MARK == "begin" )
! 215: NOMISOLATE = length(Y[0]);
! 216: RES = append(RES,Y[1]);
! 217: }
! 218: if ( MARK == "begin" ) {
! 219: F = GF; /* input polynomial -> G-base of it */
! 220: }
! 221: DEC = append(DEC,D);
! 222: if ( PRINTLOG ) {
! 223: print("");
! 224: print("# number of remainig components = ",0); print(length(RES));
! 225: }
! 226: if ( !length(RES) )
! 227: return DEC;
! 228: if ( !REDUNDANT ) { /* check whether Id(F,RES) is redundant or not */
! 229: for(L = [H],I = length(D)-1; I>=0; I--)
! 230: L=append(L,[D[I][0]]);
! 231: H = idealsetcap(L,VL,ORD); /* new intersection H */
! 232: if ( idealinc(H,F,VL,ORD) ) {
! 233: if ( PRINTLOG ) {
! 234: print("");
! 235: print("all primary components are found!");
! 236: }
! 237: return DEC;
! 238: }
! 239: REMS = mksepext(RES,H,VL); /* new remainig comps. */
! 240: } else {
! 241: REMS = mksepext(RES,H,VL); /* new rmaining comps. */
! 242: }
! 243: D = primadec_main(F,REMS,H,VL);
! 244: return append(DEC,D);
! 245: }
! 246:
! 247: /* isolated components and embedded components */
! 248: /* GF is the G-base of F, RA is the radical of F */
! 249: def isolated(IP,F,GF,RA,H,SC,VL)
! 250: {
! 251: T0 = newvect(4,time());
! 252: if ( RA[0] == "begin" )
! 253: PD = primedec_main([RA[1]],VL);
! 254: else if ( RA[0] == "ext" || RA[0] == "sep" ) {
! 255: if ( RA[0] == "sep" )
! 256: T = prime_irred(idealsav(RA[1]),VL);
! 257: else
! 258: T = [RA[1]];
! 259: if ( !NOSPECIALDEC )
! 260: PD = primedec_special(T,VL);
! 261: else
! 262: PD = primedec_main(T,VL);
! 263: } else debug;
! 264: T1 = newvect(4,time())-T0;
! 265: if ( RA[0] == "begin" ) ISORADDEC = T1;
! 266: NOMRADDEC++; RADDECTIME += T1;
! 267: if ( PRINTLOG ) {
! 268: print("number of prime components = ",0); print(length(PD),0);
! 269: print(", time = ",0); print(T1[0],0);
! 270: if ( T1[1] ) { print(" + gc : ",0); print(T1[1]); } else print("");
! 271: if ( PRINTLOG > 0 ) {
! 272: print("dimensions of primes = ",0);
! 273: for (I = 0; I < length(PD); I++) {
! 274: print(length(VL)-length(minalgdep(PD[I],VL,PRIMEORD)),0);
! 275: print(", ",0);
! 276: }
! 277: print("");
! 278: }
! 279: }
! 280: if ( RA[0] == "begin" ) { /* isolated part of initial ideal */
! 281: if ( PRINTLOG ) {
! 282: print("check 'prime decomposition = primary decomposition?'");
! 283: }
! 284: CP = idealsetcap(PD,VL,PRIMAORD);
! 285: if ( idealinc(CP,GF,VL,PRIMAORD) ) {
! 286: if ( PRINTLOG ) {
! 287: print("lucky!");
! 288: }
! 289: for (L = [],I = length(PD)-1; I >= 0; I--)
! 290: L = cons([PD[I],PD[I]],L);
! 291: return [L,[]];
! 292: }
! 293: if ( PRINTLOG ) {
! 294: print("done.");
! 295: }
! 296: }
! 297:
! 298: R = pseudo_extract(IP,F,GF,PD,H,SC,VL);
! 299:
! 300: return R;
! 301: }
! 302:
! 303: def pseudo_extract(IP,F,GF,PD,H,SC,VL)
! 304: {
! 305: REMS = DEC = PDEC = SEL = RES = [];
! 306: ZERODIM = 1; CAP = H;
! 307: for (I = 0; I < length(PD); I++) {
! 308: P = PD[I];
! 309: if ( length(minalgdep(P,VL,PRIMEORD)) != length(VL) )
! 310: ZERODIM=0;
! 311: if ( length(PD) == 1 ) { /* pseudo primary ideal case */
! 312: if ( PRINTLOG >= 1 ) { print(""); print("pseudo primary ideal"); }
! 313: DD = GF; SEL = [SC];
! 314: } else {
! 315: T0 = time();
! 316: Y = pseudodec_main(F,P,PD,VL);
! 317: T1 = time();
! 318: DD = Y[0]; SS = Y[1]; SEL = append(SEL,[SS]);
! 319: PDEC = append(PDEC,[[DD,P]]);
! 320: if ( PRINTLOG >= 1 ) {
! 321: print(""); print("pseudo primary component of ",0);
! 322: print(I,0); print(", time = ",0); print(T1[0]-T0[0]);
! 323: if ( PRINTLOG >= 4 ) { print(" = ",0); print(DD); }
! 324: }
! 325: if ( NOEMBEDDED )
! 326: continue;
! 327: }
! 328: if ( !REDUNDANT && H != [] ) { /* elim. pseu-comp. */
! 329: if ( !sepcond_ps(P,SC,VL) )
! 330: continue;
! 331: LG = localize(DD,SC,VL,PRIMAORD);
! 332: if ( idealinc(H,LG,VL,PRIMAORD)) {
! 333: if ( PRINTLOG ) {
! 334: print("eliminate the pseudo primary component ",0);
! 335: print(I);
! 336: }
! 337: continue;
! 338: }
! 339: }
! 340: if ( PRINTLOG ) {
! 341: print("extraction of the pseudo primary component ",0);
! 342: print(I);
! 343: if ( PRINTLOG > 2 ) {
! 344: print(" associated prime of pseudo primary ideal = ",0);
! 345: print(P);
! 346: }
! 347: }
! 348: U = extraction(DD,P,VL);
! 349: if ( !REDUNDANT && H != [] && idealinc(H,U[0][0],VL,PRIMAORD) ) {
! 350: if ( PRINTLOG )
! 351: print("redundant primary component!");
! 352: } else {
! 353: DEC = append(DEC,[U[0]]);
! 354: H = idealcap(H,U[0][0],VL,PRIMAORD);
! 355: if ( idealeq(IP,H) ) {
! 356: if ( PRINTLOG ) {
! 357: print("");
! 358: print("all primary components are found!");
! 359: }
! 360: return [DEC,[]];
! 361: }
! 362: }
! 363: if ( !ISOLATED && U[1] != [] )
! 364: if ( sepcond_re(U[1],SC,VL) ) {
! 365: NSC = setunion([SS,SC]); /* new separating condition */
! 366: REM = cons(DD,append(U[1],[NSC,H]));
! 367: REMS = append(REMS,[REM]);
! 368: }
! 369: }
! 370: if ( NOEMBEDDED )
! 371: DEC = PDEC;
! 372: if ( length(PD) != 1 && !NOEMBEDDED && !ISOLATED && !ZERODIM ) {
! 373: for (QD=[],I=length(PDEC)-1;I>=0;I--)
! 374: QD = cons(PDEC[I][0],QD);
! 375: RES = ["sep",PD,QD,GF];
! 376: if ( sepcond_re(append(RES,[SEL]),SC,VL) ) {
! 377: REM = cons(F,append(RES,[SC,H]));
! 378: REMS = append(REMS,[REM]);
! 379: }
! 380: }
! 381: return [DEC,REMS];
! 382: }
! 383:
! 384: /* F:input set, PD:primes of radical, E:higher dimensional ideal */
! 385: /* length(PD) > 1 */
! 386: /* output : pseudo primary comp., remaining comp., selectors */
! 387: def pseudodec_main(F,P,PD,VL)
! 388: {
! 389: ZERODIM = 1;
! 390: S = selects(P,PD,VL,SELECTFLAG);
! 391: R = localize(F,S,VL,PRIMAORD);
! 392: if ( R[0] == 1 || R[0] == -1 ) {
! 393: print("error, R = ",0); print(R);
! 394: debug;
! 395: }
! 396: R = idealnormal(R);
! 397: return [R,S];
! 398: }
! 399:
! 400: /* Id(GF) is a pseudo primary ideal. (GF must be the G-base.) */
! 401: def extraction(GF,Pr,VL)
! 402: {
! 403: TMPORD1=TMPORD2=0;
! 404: V = minalgdep(Pr,VL,PRIMEORD);
! 405: U = listminus(VL,V);
! 406: V0 = append(V,U);
! 407: if ( V0 != VL ) {
! 408: ORD = [[TMPORD1,length(V)],[TMPORD2,length(U)]];
! 409: GR(G,GF,V0,ORD);
! 410: } else
! 411: G = GF;
! 412: dp_ord(TMPORD1);
! 413: for (LL = [],HC = 1,I = 0; I < length(G); I++)
! 414: LL = append(LL,cdr(fctr(dp_hc(dp_ptod(G[I],V)))));
! 415: for (L=[],I=0;I<length(LL);I++)
! 416: L = cons(LL[I][0],L);
! 417: L = setunion([L]);
! 418: for (S=1,SL=[],I=0;I<length(L);I++) {
! 419: S *= L[I];
! 420: if ( SELECTFLAG )
! 421: SL = cons(L[I],SL);
! 422: }
! 423: if ( !SELECTFLAG )
! 424: SL= [S];
! 425: if ( PRINTLOG > 1 ) {
! 426: print("extractor = ",0);
! 427: print(S);
! 428: }
! 429: T0 = time()[0];
! 430: Q = localize(GF,SL,VL,PRIMAORD);
! 431: Q = idealnormal(Q);
! 432: DEC = [Q,Pr];
! 433: if ( PRINTLOG ) {
! 434: print("find a primary component! time = ",0);
! 435: print(time()[0]-T0);
! 436: if (PRINTLOG >= 3){
! 437: print(" associated prime of primary component = ",0);
! 438: print(DEC[1]);
! 439: }
! 440: if (PRINTLOG >= 4){print(" primary component = ",0); print(Q);}
! 441: }
! 442: if ( !ISOLATED && !NOEMBEDDED && SL != [1]
! 443: && length(V) != length(VL) /* nonzerodim */
! 444: && (REDUNDANT || !idealinc(Q,GF,VL,PRIMAORD)) ) {
! 445: REM = ["ext",[Q,Pr],GF,S];
! 446: if ( PRINTLOG ) {
! 447: print("find the remaining component of the extraction");
! 448: }
! 449: } else {
! 450: REM = [];
! 451: if ( PRINTLOG ) {
! 452: print("eliminate the remaining component of the extraction");
! 453: }
! 454: }
! 455: return [DEC,REM];
! 456: }
! 457:
! 458: /* sep. cond. for pseudo-primary components */
! 459: def sepcond_ps(P,SC,VL)
! 460: {
! 461: for (J = 0; J < length(SC); J++) {
! 462: if ( idealinc([SC[J]],P,VL,PRIMEORD) )
! 463: break; /* separating condition */
! 464: }
! 465: if ( J != length(SC) ) {
! 466: if ( PRINTLOG ) {
! 467: print("");
! 468: print("elim. the pseudo primary comp. by separating cond.");
! 469: }
! 470: return 0;
! 471: }
! 472: return 1;
! 473: }
! 474:
! 475: /* sep. cond. for rem. components. */
! 476: /* REM = ["ext",[Q,Pr],GF,S] or ["sep",PD,QD,GF,SEL], SC : sep.cond. */
! 477: def sepcond_re(REM,SC,VL)
! 478: {
! 479: for (S=1,I=0;I<length(SC);I++)
! 480: S *= SC[I];
! 481: if (REM[0] == "ext") {
! 482: F = cons(REM[3],REM[1][1]);
! 483: L = localize(F,[S],VL,PRIMAORD);
! 484: if ( L != [1] )
! 485: return 1;
! 486: else
! 487: return 0;
! 488: } else if (REM[0] == "sep") {
! 489: PL = REM[1]; SEL = REM[4];
! 490: for (I=0;I<length(PL);I++) {
! 491: F = append(SEL[I],PL[I]);
! 492: L = localize(F,[S],VL,PRIMAORD);
! 493: if ( L != [1] )
! 494: return 1;
! 495: }
! 496: return 0;
! 497: }
! 498: }
! 499:
! 500: def minalgdep(Pr,VL,ORD)
! 501: {
! 502: T0=newvect(4,time());
! 503: if (MAXALGIND)
! 504: R = minalgdep1(Pr,VL,ORD); /* M.I.S. */
! 505: else
! 506: R = minalgdep0(Pr,VL,ORD); /* M.S.I.S. */
! 507: T1=newvect(4,time())-T0; MISTIME += T1;
! 508: if ( PRINTLOG >= 6 ) {
! 509: if (MAXALGIND) print("M.I.S. time = ",0);
! 510: else print("M.S.I.S. time = ",0);
! 511: print(T1[0]);
! 512: }
! 513: return R;
! 514: }
! 515:
! 516: def minalgdep0(Pr,VL,ORD)
! 517: {
! 518: dp_ord(ORD); N = length(Pr);
! 519: for (HD = [],I = N-1; I >= 0; I--)
! 520: HD = cons(vars(dp_dtop(dp_ht(dp_ptod(Pr[I],VL)),VL)),HD);
! 521: for (R = X = [], I = length(VL)-1; I >= 0; I--) {
! 522: V = VL[I];
! 523: TX = cons(V,X);
! 524: for (J = 0; J < N; J++) {
! 525: if ( varinc(HD[J],TX) )
! 526: break;
! 527: }
! 528: if ( J == N )
! 529: X = TX;
! 530: else
! 531: R = cons(V,R);
! 532: }
! 533: return R;
! 534: }
! 535:
! 536: def minalgdep1(Pr,VL,ORD)
! 537: {
! 538: R = all_msis(Pr,VL,ORD);
! 539: for (E=I=0,D=length(R[0]);I<length(R);I++) {
! 540: if ( length(R[I])>=D ) {
! 541: D=length(R[I]); E=I;
! 542: }
! 543: }
! 544: S = listminus(VL,R[E]);
! 545: return S;
! 546: }
! 547:
! 548: def all_msis(Pr,VL,ORD)
! 549: {
! 550: dp_ord(ORD); N = length(Pr);
! 551: for (HD = [],I = N-1; I >= 0; I--)
! 552: HD = cons(vars(dp_dtop(dp_ht(dp_ptod(Pr[I],VL)),VL)),HD);
! 553: R = dimrec(HD,0,[],[],VL);
! 554: return R;
! 555: }
! 556:
! 557: def dimrec(S,K,U,M,X)
! 558: {
! 559: MM = M;
! 560: for (I=K;I<length(X);I++) {
! 561: TX = append(U,[X[I]]);
! 562: for (J = 0; J < length(S); J++) {
! 563: if ( varinc(S[J],TX) )
! 564: break;
! 565: }
! 566: if ( J == length(S) )
! 567: MM = dimrec(S,I+1,TX,MM,X);
! 568: }
! 569: for (J = 0; J < length(MM); J++) {
! 570: if ( varinc(U,MM[J]) )
! 571: break;
! 572: }
! 573: if ( J == length(MM) )
! 574: MM = append(MM,[U]);
! 575: return MM;
! 576: }
! 577:
! 578: /* if V is min.alg.dep. return 1 */
! 579: def ismad(Pr,V,VL,ORD)
! 580: {
! 581: dp_ord(ORD); N = length(Pr);
! 582: for (HD = [],I = N-1; I >= 0; I--)
! 583: HD = cons(vars(dp_dtop(dp_ht(dp_ptod(Pr[I],VL)),VL)),HD);
! 584: W = listminus(VL,V); L = append(W,V);
! 585: for (R = TX = [],I = 0; I < length(L); I++) {
! 586: TX = cons(L[I],TX);
! 587: for (J = 0; J < N; J++) {
! 588: if ( varinc(HD[J],TX) )
! 589: break;
! 590: }
! 591: if ((I<length(W)&&J!=N) || (I>=length(W)&&J==N))
! 592: return 0;
! 593:
! 594: }
! 595: return 1;
! 596: }
! 597:
! 598: /* output:list of [flag, generator, primes, sep.cond., intersection] */
! 599: def mksepext(RES,H,VL)
! 600: {
! 601: for (R = [],I=length(RES)-1;I>=0;I--) {
! 602: W = RES[I];
! 603: if ( COMPLETE || EXTENDED )
! 604: HI = idealcap(H,W[6],VL,PRIMAORD);
! 605: else
! 606: HI = H;
! 607: if (W[1] == "sep") {
! 608: E = mksep(W[2],W[3],W[4],VL);
! 609: F = append(E[0],W[0]);
! 610: if ( ( COMPLETE || EXTENDED ) && !REDUNDANT )
! 611: HI = idealsetcap(cons(HI,W[3]),VL,PRIMAORD);
! 612: R = cons(["sep",F,E[1],W[5],HI],R);
! 613: } else if (W[1] == "ext") {
! 614: E = mkext(W[2][0],W[3],W[4],VL);
! 615: F = cons(E[0],W[0]);
! 616: P = cons(E[1],W[2][1]); /* prime component of rem. comp. */
! 617: R = cons(["ext",F,P,W[5],HI],R);
! 618: } else debug;
! 619: }
! 620: return R;
! 621: }
! 622:
! 623: /* make extractor with exponents ; F must be G-Base */
! 624: /* output : extractor, max.sqfr.of extractor */
! 625: def mkext(Q,F,S,VL)
! 626: {
! 627: if ( PRINTLOG > 1 ) {
! 628: print("make extractor = ",0);
! 629: }
! 630: if ( S == 1 || S == -1 ) {
! 631: print(1); return [1,1];
! 632: }
! 633: DD=dp_mkdist([Q],F,VL,PRIMAORD);
! 634: DQ=DD[0][0]; DF=DD[1];
! 635: for (L=fctr(S),SL=[],J=length(L)-1;J>0;J--)
! 636: SL=cons(dp_ptod(L[J][0],VL),SL);
! 637: K = dp_fexponent(dp_ptod(S,VL),DF,DQ);
! 638: E = dp_vectexp(SL,EX=dp_minexp(K,SL,DF,DQ));
! 639: if ( E != 1 ) E = dp_dtop(E,VL);
! 640: for (U=1, I=0;I<size(EX)[0];I++)
! 641: if ( EX[I] )
! 642: U *= L[I+1][0];
! 643: /* for (L=sqfr(E),U=1,J=length(L)-1;J>0;J--)
! 644: U *= L[J][0]; */
! 645: if ( PRINTLOG > 1 ) {
! 646: print(E);
! 647: }
! 648: return [E,U];
! 649: }
! 650:
! 651: /* make separators with exponents ; F must be G-Base */
! 652: /* output [separator, list of components of radical] */
! 653: def mksep(PP,QQ,F,VL)
! 654: {
! 655: if ( PRINTLOG > 1 ) {
! 656: print("make separators, ",0); print(length(PP));
! 657: }
! 658: DD=dp_mkdist(QQ,F,VL,PRIMAORD);
! 659: DQ=DD[0]; DF=DD[1];
! 660: E = dp_mksep(PP,DQ,DF,VL,PRIMAORD);
! 661: for (RA=[],I=length(E)-1;I>=0;I--) {
! 662: for (L=fctr(E[I]),J=length(L)-1;J>0;J--) {
! 663: ES=cons(L[J][0],PP[I]);
! 664: RA = cons(ES,RA); /* radical of remaining comp. */
! 665: }
! 666: }
! 667: return [E,RA];
! 668: }
! 669:
! 670: def dp_mksep(PP,DQ,DF,VL,ORD)
! 671: {
! 672: dp_ord(ORD);
! 673: for (E=[],I=0;I<length(PP);I++) {
! 674: T0=time()[0];
! 675: if ( CONTINUE ) {
! 676: E = bload("sepa");
! 677: I = length(E);
! 678: CONTINUE = 0;
! 679: }
! 680: S = selects(PP[I],PP,VL,0)[0];
! 681: for (L=fctr(S),SL=[],J=length(L)-1;J>0;J--)
! 682: SL=cons(dp_ptod(L[J][0],VL),SL);
! 683: K = dp_fexponent(dp_ptod(S,VL),DF,DQ[I]);
! 684: M = dp_vectexp(SL,dp_minexp(K,SL,DF,DQ[I]));
! 685: if (M != 1) M = dp_dtop(M,VL);
! 686: E = append(E,[M]);
! 687: if ( PRINTLOG > 1 ) {
! 688: print("separator of ",0); print(I,0);
! 689: print(" in ",0); print(length(PP),0);
! 690: print(", time = ",0); print(time()[0]-T0,0); print(", ",0);
! 691: printsep(cdr(fctr(M))); print("");
! 692: }
! 693: if (BSAVE) {
! 694: bsave(E,"sepa");
! 695: print("bsave to sepa");
! 696: }
! 697: }
! 698: return E;
! 699: }
! 700:
! 701: extern AFO$
! 702:
! 703: /* F,Id,Q is given by distributed polynomials. Id is a G-Base.*/
! 704: /* get the exponent K s.t. (Id : F^K) = Q, Id is a G-Basis */
! 705: def dp_fexponent(F,Id,Q)
! 706: {
! 707: if (!AFO)
! 708: for (IN=[],I=size(Id)[0]-1;I>=0;I--) IN = cons(I,IN);
! 709: else
! 710: for (IN=[],I=0;I<size(Id)[0];I++) IN = cons(I,IN);
! 711: NF(G,IN,F,Id,0);
! 712: if ( F != G ) { F = G; }
! 713: N=size(Q)[0]; U = newvect(N);
! 714: for (I=0;I<N;I++)
! 715: U[I]=Q[I];
! 716: for (K = 1;; K++) { /* U[I] = Q[I]*F^K mod Id */
! 717: for (FLAG = I = 0; I < N; I++) {
! 718: NF(U[I],IN,U[I]*F,Id,1);
! 719: if ( U[I] ) FLAG = 1;
! 720: }
! 721: if ( !FLAG )
! 722: return K;
! 723: }
! 724: }
! 725:
! 726: /* F must be a G-Base. SL is a set of factors of S.*/
! 727: def dp_minexp(K,SL,F,Q)
! 728: {
! 729: if (!AFO)
! 730: for (IN=[],I=size(F)[0]-1;I>=0;I--) IN = cons(I,IN);
! 731: else
! 732: for (IN=[],I=0;I<size(F)[0];I++) IN = cons(I,IN);
! 733: N = length(SL); M = size(Q)[0];
! 734: EX = newvect(N); U = newvect(M); T = newvect(M);
! 735: for (I=0;I<N;I++)
! 736: EX[I]=K;
! 737: for (I=0;I<M;I++) {
! 738: NF(Q[I],IN,Q[I],F,1);
! 739: }
! 740: for (I=0;I<N;I++) {
! 741: EX[I] = 0;
! 742: for (FF=1,II=0;II<N;II++)
! 743: for (J = 0; J < EX[II]; J++) {
! 744: NF(FF,IN,FF*SL[II],F,1);
! 745: }
! 746: for (J = 0; J < M; J++)
! 747: U[J] = Q[J]*FF;
! 748: for (; EX[I] < K; EX[I]++) {
! 749: FLAG = 0;
! 750: for (J = 0; J < M; J++) {
! 751: NF(U[J],IN,U[J],F,1);
! 752: if ( U[J] ) FLAG = 1;
! 753: }
! 754: if ( !FLAG )
! 755: break;
! 756: if ( EX[I] < K-1 )
! 757: for (J = 0; J < M; J++) U[J] = U[J]*SL[I];
! 758: }
! 759: }
! 760: return EX;
! 761: }
! 762:
! 763: def dp_vectexp(SS,EE)
! 764: {
! 765: N = length(SS);
! 766: for (A=1,I=0;I<N;I++)
! 767: for (J = 0; J < EE[I]; J++)
! 768: A *= SS[I];
! 769: return A;
! 770: }
! 771:
! 772: def dp_mkdist(QQ,F,VL,ORD)
! 773: {
! 774: dp_ord(ORD);
! 775: for (DQ=[],I=length(QQ)-1;I>=0;I--) {
! 776: for (T=[],J=length(QQ[I])-1;J>=0;J--)
! 777: T = cons(dp_ptod(QQ[I][J],VL),T);
! 778: DQ=cons(newvect(length(T),T),DQ);
! 779: }
! 780: for (T=[],J=length(F)-1;J>=0;J--)
! 781: T = cons(dp_ptod(F[J],VL),T);
! 782: DF = newvect(length(T),T);
! 783: return [DQ,DF]$
! 784: }
! 785:
! 786: /* if FLAG == 0 return singltonset */
! 787: def selects(P,PD,VL,FLAG)
! 788: {
! 789: PP=dp_mkdist([],P,VL,PRIMEORD)[1];
! 790: for (M=[],I=length(P)-1;I>=0;I--)
! 791: M = cons(I,M);
! 792: for (SL = [],I = length(PD)-1; I >= 0; I--) {
! 793: B = PD[I];
! 794: if (B == P) continue;
! 795: for (L = [],J = 0; J < length(B); J++) {
! 796: A = B[J];
! 797: NF(E,M,dp_ptod(A,VL),PP,0);
! 798: if ( E ) {
! 799: L = append(L,[A]); /* A \notin Id(P) */
! 800: }
! 801: }
! 802: SL = cons(L,SL); /* candidate of separators */
! 803: }
! 804: for (S=[],I=length(SL)-1;I>=0;I--) {/* choose an element from L */
! 805: if ( FLAG >= 2 ) {
! 806: S = append(SL[I],S);
! 807: continue;
! 808: }
! 809: C = minselect(SL[I],VL,PRIMAORD);
! 810: S = cons(C,S);
! 811: }
! 812: S = setunion([S]);
! 813: if ( FLAG == 3 || length(S) == 1 )
! 814: return S;
! 815: if ( FLAG == 2 ) {
! 816: for (R=1,I=0;I<length(S);I++)
! 817: R *= S[I];
! 818: return [R];
! 819: }
! 820: /* FLAG == 0 || FLAG == 1 */
! 821: for (T=[]; S!=[];S = cdr(S)) { /* FLAG <= 1 and length(S) != 1 */
! 822: A = car(S);
! 823: U = append(T,cdr(S));
! 824: for (R=1,I=0;I<length(U);I++)
! 825: R *= U[I];
! 826: for (I=0;I<length(PD);I++) {
! 827: if ( PD[I] != P && !idealinc([R],PD[I],VL,PRIMAORD) )
! 828: break;
! 829: }
! 830: if ( I != length(PD) )
! 831: T = append(T,[A]);
! 832: }
! 833: if ( FLAG )
! 834: return T;
! 835: for (R=1,I=0;I<length(T);I++)
! 836: R *= T[I];
! 837: return [R]; /* if FLAG == 0 */
! 838: }
! 839:
! 840: def minselect(L,VL,ORD)
! 841: {
! 842: dp_ord(ORD);
! 843: F = dp_ptod(L[0],VL); MAG=dp_mag(F); DEG=dp_td(F);
! 844: for (J = 0, I = 1; I < length(L); I++) {
! 845: A = dp_ptod(L[I],VL); M=dp_mag(A); D=dp_td(A);
! 846: if ( dp_ht(A) > dp_ht(F) )
! 847: continue;
! 848: else if ( dp_ht(A) == dp_ht(F) && (D > DEG || (D == DEG && M > MAG)))
! 849: continue;
! 850: F = A; J = I; MAG = M; DEG=D;
! 851: }
! 852: return L[J];
! 853: }
! 854:
! 855: /* localization Id(F)_MI \cap Q[VL] */
! 856: /* output is the G-base w.r.t ordering O */
! 857: def localize(F,MI,VL,O)
! 858: {
! 859: if ( MI == [1] || MI == [-1] )
! 860: return F;
! 861: for (SVL = [],R = [],I = 0; I < length(MI); I++) {
! 862: V = strtov("zz"+rtostr(I));
! 863: R = append(R,[V*MI[I]-1]);
! 864: SVL = append(SVL,[V]);
! 865: }
! 866: if ( O == 0 ) {
! 867: dp_nelim(length(SVL)); ORD = 6;
! 868: } else if ( O == 1 || O == 2 ) {
! 869: ORD = [[0,length(SVL)],[O,length(VL)]];
! 870: } else if ( O == 3 || O == 4 || O == 5 )
! 871: ORD = [[0,length(SVL)],[O-3,length(VL)-1],[2,1]];
! 872: else
! 873: error("invarid ordering");
! 874: GR(G,append(F,R),append(SVL,VL),ORD);
! 875: S = varminus(G,SVL);
! 876: return S;
! 877: }
! 878:
! 879: def varminus(G,VL)
! 880: {
! 881: for (S = [],I = 0; I < length(G); I++) {
! 882: V = vars(G[I]);
! 883: if (listminus(V,VL) == V)
! 884: S = append(S,[G[I]]);
! 885: }
! 886: return S;
! 887: }
! 888:
! 889: def idealnormal(L)
! 890: {
! 891: R=[];
! 892: for (I=length(L)-1;I>=0;I--) {
! 893: A = ptozp(L[I]);
! 894: V = vars(A);
! 895: for (B = A,J = 0;J < length(V);J++)
! 896: B = coef(B,deg(B,V[J]),V[J]);
! 897: R = cons((B>0?A:-A),R);
! 898: }
! 899: return R;
! 900: }
! 901:
! 902: def idealsav(L) /* sub procedure of welldec and normposdec */
! 903: {
! 904: if ( PRINTLOG >= 4 )
! 905: print("multiple check ",2);
! 906: for (R = [],I = 0,N=length(L); I < N; I++) {
! 907: if ( PRINTLOG >= 4 && !irem(I,10) )
! 908: print(".",2);
! 909: if ( R == [] )
! 910: R = append(R,[L[I]]);
! 911: else {
! 912: for (J = 0; J < length(R); J++)
! 913: if ( idealeq(L[I],R[J]) )
! 914: break;
! 915: if ( J == length(R) )
! 916: R = append(R,[L[I]]);
! 917: }
! 918: }
! 919: if ( PRINTLOG >= 4 )
! 920: print("done.");
! 921: return R;
! 922: }
! 923:
! 924: /* intersection of ideals in PL, PL : list of ideals */
! 925: /* VL : variable list, O : output the G-base w.r.t order O */
! 926: def idealsetcap(PL,VL,O)
! 927: {
! 928: for (U = [], I = 0; I < length(PL); I++)
! 929: if ( PL[I] != [] )
! 930: U = append(U,[PL[I]]);
! 931: if ( U == [] )
! 932: return [];
! 933: if ( PRINTLOG >= 4 ) {print("intersection of ideal ",0); print(length(U),0);}
! 934: for (L = U[0],I=1;I<length(U);I++) {
! 935: if ( PRINTLOG >=4 ) print(".",2);
! 936: L = idealcap(L,U[I],VL,O);
! 937: }
! 938: if ( PRINTLOG >=4 ) print("");
! 939: return L;
! 940: }
! 941:
! 942: /* return intersection set of ideals P and Q */
! 943: def idealcap(P,Q,VL,O)
! 944: {
! 945: if ( P == [] )
! 946: if ( VL )
! 947: return Q;
! 948: else
! 949: return [];
! 950: if ( Q == [] )
! 951: return P;
! 952: T = tmp;
! 953: for (PP = [],I = 0; I < length(P); I++)
! 954: PP = append(PP,[P[I]*T]);
! 955: for (QQ = [],I = 0; I < length(Q); I++)
! 956: QQ = append(QQ,[Q[I]*(T-1)]);
! 957: if ( O == 0 ) {
! 958: dp_nelim(1); ORD = 6;
! 959: } else if ( O == 1 || O == 2 )
! 960: ORD = [[2,1],[O,length(VL)]];
! 961: else if ( O == 3 || O == 4 || O == 5 )
! 962: ORD = [[2,1],[O-3,length(VL)-1],[2,1]];
! 963: else
! 964: error("invarid ordering");
! 965: GR(G,append(PP,QQ),append([T],VL),ORD);
! 966: S = varminus(G,[T]);
! 967: return S;
! 968: }
! 969:
! 970: /* return ideal P : Q */
! 971: def idealquo(P,Q,VL,O)
! 972: {
! 973: for (R = [], I = 0; I < length(Q); I++) {
! 974: F = car(Q);
! 975: G = idealcap(P,[F],VL,O);
! 976: for (H = [],J = 0; J < length(G); J++) {
! 977: H = append(H,[sdiv(G[J],F)]);
! 978: }
! 979: R = idealcap(R,H,VL,O);
! 980: }
! 981: }
! 982:
! 983: /* if ideal Q includes P then return 1 else return 0 */
! 984: /* Q must be a Groebner Basis w.r.t ordering ORD */
! 985: def idealinc(P,Q,VL,ORD)
! 986: {
! 987: dp_ord(ORD);
! 988: for (T=IN=[],I=length(Q)-1;I>=0;I--) {
! 989: T=cons(dp_ptod(Q[I],VL),T);
! 990: IN=cons(I,IN);
! 991: }
! 992: DQ = newvect(length(Q),T);
! 993: for (I = 0; I < length(P); I++) {
! 994: NF(E,IN,dp_ptod(P[I],VL),DQ,0);
! 995: if ( E )
! 996: return 0;
! 997: }
! 998: return 1;
! 999: }
! 1000:
! 1001: /* FL : list of polynomial set */
! 1002: def primedec_main(FL,VL)
! 1003: {
! 1004: if ( PRINTLOG ) {
! 1005: print("prime decomposition of the radical");
! 1006: }
! 1007: if ( PRINTLOG >= 2 )
! 1008: print("G-base factorization");
! 1009: PP = gr_fctr(FL,VL);
! 1010: if ( PRINTLOG >= 2 )
! 1011: print("irreducible variety decomposition");
! 1012: RP = welldec(PP,VL);
! 1013: SP = normposdec(RP,VL);
! 1014: for (L=[],I=length(SP)-1;I>=0;I--) {
! 1015: L=cons(idealnormal(SP[I]),L); /* head coef. > 0 */
! 1016: }
! 1017: SP = L;
! 1018: return SP;
! 1019: }
! 1020:
! 1021: def gr_fctr(FL,VL)
! 1022: {
! 1023: for (TP = [],I = 0; I<length(FL); I++ ) {
! 1024: F = FL[I];
! 1025: SF = idealsqfr(F);
! 1026: if ( !idealeq(F,SF) ) { GR(F,SF,VL,PRIMEORD); }
! 1027: DIRECTRY = []; COMMONIDEAL=[1];
! 1028: SP = gr_fctr_sub(F,VL);
! 1029: TP = append(TP,SP);
! 1030: }
! 1031: TP = idealsav(TP);
! 1032: TP = prime_irred(TP,VL);
! 1033: return TP;
! 1034: }
! 1035:
! 1036: def gr_fctr_sub(G,VL)
! 1037: {
! 1038: CONTINUE;
! 1039: if ( length(G) == 1 && type(G[0]) == 1 )
! 1040: return [G];
! 1041: RL = [];
! 1042: for (I = 0; I < length(G); I++) {
! 1043: FL = fctr(G[I]); L = length(FL); N = FL[1][1];
! 1044: if (L > 2 || N > 1) {
! 1045: TLL = [];
! 1046: for (J = 1; J < L; J++) {
! 1047: if ( CONTINUE ) {
! 1048: JCOPP=bload("jcopp");
! 1049: J = JCOPP[0]+1;
! 1050: COMMONIDEAL = JCOPP[1];
! 1051: RL = JCOPP[2];
! 1052: CONTINUE = 0;
! 1053: }
! 1054: if ( PRINTLOG >= 5 ) {
! 1055: for (D = length(DIRECTRY)-1; D >= 0; D--)
! 1056: print(DIRECTRY[D],0);
! 1057: print([L-1,J,length(RL)]);
! 1058: /*
! 1059: L-1:a number of factors of polynomial
! 1060: J:a factor executed now [0,...L-1]
! 1061: length(RL):a number of prime components
! 1062: */
! 1063: }
! 1064: W = cons(FL[J][0],G);
! 1065: GR(NG,W,VL,PRIMEORD);
! 1066: TNG = idealsqfr(NG);
! 1067: if ( NG != TNG ) { GR(NG,TNG,VL,PRIMEORD); }
! 1068: if ( idealinc(COMMONIDEAL,NG,VL,PRIMEORD) )
! 1069: continue;
! 1070: else {
! 1071: DIRECTRY=cons(L-J-1,DIRECTRY);
! 1072: DG = gr_fctr_sub(NG,VL);
! 1073: DIRECTRY=cdr(DIRECTRY);
! 1074: RL = append(RL,DG);
! 1075: if ( J <= L-2 && !idealinc(COMMONIDEAL,NG,VL,PRIMEORD)
! 1076: && COMMONCHECK ) {
! 1077: if ( PRINTLOG >= 5 ) {
! 1078: for (D = 0; D < length(DIRECTRY); D++) print(" ",2);
! 1079: print("intersection ideal");
! 1080: }
! 1081: COMMONIDEAL=idealcap(COMMONIDEAL,NG,VL,PRIMEORD);
! 1082: }
! 1083: }
! 1084: if ( BSAVE && !length(DIRECTRY) ) {
! 1085: bsave([J,COMMONIDEAL,RL],"jcopp");
! 1086: if ( PRINTLOG >= 2 )
! 1087: print("bsave j, intersection ideal and primes to icopp ");
! 1088: }
! 1089: }
! 1090: break;
! 1091: }
! 1092: }
! 1093: if (I == length(G)) {
! 1094: RL = append([G],RL);
! 1095: if ( PRINTLOG >= 5 ) {
! 1096: for (D = 0; D < length(DIRECTRY)-1; D++) print(" ",0);
! 1097: print("prime G-base ",0);
! 1098: if ( PRINTLOG >= 6 )
! 1099: print(G);
! 1100: else
! 1101: print("");
! 1102: }
! 1103: }
! 1104: return RL;
! 1105: }
! 1106:
! 1107: def prime_irred(TP,VL)
! 1108: {
! 1109: if ( PRINTLOG >= 4 )
! 1110: print("irredundancy check for prime ideal");
! 1111: N = length(TP);
! 1112: for (P = [], I = 0; I < N; I++) {
! 1113: if ( PRINTLOG >= 4 )
! 1114: print(".",2);
! 1115: for (J = 0; J < N; J++)
! 1116: if ( I != J && idealinc(TP[J],TP[I],VL,PRIMEORD) )
! 1117: break;
! 1118: if (J == N)
! 1119: P = append(P,[TP[I]]);
! 1120: }
! 1121: if ( PRINTLOG >= 4 )
! 1122: print("done.");
! 1123: return P;
! 1124: }
! 1125:
! 1126: /* if V1 \subset V2 then return 1 else return 0 */
! 1127: def varinc(V1,V2)
! 1128: {
! 1129: N1=length(V1); N2=length(V2);
! 1130: for (I=N1-1;I>=0;I--) {
! 1131: for (J=N2-1;J>=0;J--)
! 1132: if (V1[I]==V2[J])
! 1133: break;
! 1134: if (J==-1)
! 1135: return 0;
! 1136: }
! 1137: return 1;
! 1138: }
! 1139:
! 1140: def setunion(PS)
! 1141: {
! 1142: for (R=C=[]; PS != [] && car(PS); PS=cdr(PS)) {
! 1143: for (L = car(PS); L != []; L = cdr(L)) {
! 1144: A = car(L);
! 1145: for (G = C; G != []; G = cdr(G)) {
! 1146: if ( A == car(G) || -A == car(G) )
! 1147: break;
! 1148: }
! 1149: if ( G == [] ) {
! 1150: R = append(R,[A]);
! 1151: C = cons(A,C);
! 1152: }
! 1153: }
! 1154: }
! 1155: return R;
! 1156: }
! 1157:
! 1158: def idealeq(L,M)
! 1159: {
! 1160: if ((N1=length(L)) != (N2=length(M)))
! 1161: return 0;
! 1162: for (I = 0; I < length(L); I++) {
! 1163: for (J = 0; J < length(M); J++)
! 1164: if (L[I] == M[J] || L[I] == - M[J])
! 1165: break;
! 1166: if (J == length(M))
! 1167: return 0;
! 1168: }
! 1169: return 1;
! 1170: }
! 1171:
! 1172: def listminus(L,M)
! 1173: {
! 1174: for (R = []; L != []; L = cdr(L) ) {
! 1175: A = car(L);
! 1176: for (G = M; G != []; G = cdr(G)) {
! 1177: if ( A == car(G) )
! 1178: break;
! 1179: }
! 1180: if ( G == [] ) {
! 1181: R = append(R,[A]);
! 1182: M = cons(A,M);
! 1183: }
! 1184: }
! 1185: return R;
! 1186: }
! 1187:
! 1188: def idealsqfr(G)
! 1189: {
! 1190: for(Flag=0,LL=[],I=length(G)-1;I>=0;I--) {
! 1191: for(A=1,L=sqfr(G[I]),J=1;J<length(L);J++)
! 1192: A*=L[J][0];
! 1193: LL=cons(A,LL);
! 1194: }
! 1195: return LL;
! 1196: }
! 1197:
! 1198: def welldec(PRIME,VL)
! 1199: {
! 1200: PP = PRIME; NP = [];
! 1201: while ( !Flag ) {
! 1202: LP = [];
! 1203: Flag = 1;
! 1204: for (I=0;I<length(PP);I++) {
! 1205: U = welldec_sub(PP[I],VL);
! 1206: if (length(U) >= 2) {
! 1207: Flag = 0;
! 1208: if ( PRINTLOG >= 3 ) {
! 1209: print("now decomposing to irreducible varieties ",0);
! 1210: print(I,0); print(" ",0); print(length(PP));
! 1211: }
! 1212: DIRECTRY = []; COMMONIDEAL=[1];
! 1213: PL1 = gr_fctr([U[0]],VL);
! 1214: DIRECTRY = []; COMMONIDEAL=[1];
! 1215: PL2 = gr_fctr([U[1]],VL);
! 1216: LP = append(LP,append(PL1,PL2));
! 1217: }
! 1218: else
! 1219: NP = append(NP,U);
! 1220: }
! 1221: PP = LP;
! 1222: if ( PRINTLOG > 3 ) {
! 1223: print("prime length and non-prime length = ",0);
! 1224: print(length(NP),0); print(" ",0); print(length(PP));
! 1225: }
! 1226: }
! 1227: if ( length(PRIME) != length(NP) ) {
! 1228: NP = idealsav(NP);
! 1229: NP = prime_irred(NP,VL);
! 1230: }
! 1231: return NP;
! 1232: for (I = 0; I<length(PP);I++) {
! 1233: }
! 1234: }
! 1235:
! 1236: def welldec_sub(PP,VL)
! 1237: {
! 1238: VV = minalgdep(PP,VL,PRIMEORD);
! 1239: S = wellsep(PP,VV,VL);
! 1240: if ( S == 1 )
! 1241: return [PP];
! 1242: P1 = localize(PP,[S],VL,PRIMEORD);
! 1243: if ( idealeq(P1,PP) )
! 1244: return([PP]);
! 1245: GR(P2,cons(S,PP),VL,PRIMEORD);
! 1246: return [P1,P2];
! 1247: }
! 1248:
! 1249: def wellsep(PP,VV,VL)
! 1250: {
! 1251: TMPORD = 0;
! 1252: V0 = listminus(VL,VV);
! 1253: V1 = append(VV,V0);
! 1254: /* ORD = [[TMPORD,length(VV)],[0,length(V0)]]; */
! 1255: dp_nelim(length(VV)); ORD = 6;
! 1256: GR(PP,PP,V1,ORD);
! 1257: dp_ord(TMPORD);
! 1258: for (E=1,I=0;I<length(PP);I++)
! 1259: E = lcm(E,dp_hc(dp_ptod(PP[I],VV)));
! 1260: for (F=1,L=sqfr(E),K=1;K<length(L);K++)
! 1261: F *= L[K][0];
! 1262: return F;
! 1263: }
! 1264:
! 1265: /* prime decomposition by using primitive element method */
! 1266: def normposdec(NP,VL)
! 1267: {
! 1268: if ( PRINTLOG >= 3 )
! 1269: print("radical decomposition by using normal position.");
! 1270: for (R=MP=[],I=0;I<length(NP);I++) {
! 1271: V=minalgdep(NP[I],VL,PRIMEORD);
! 1272: L=raddec(NP[I],V,VL,1);
! 1273: if ( PRINTLOG >= 3 )
! 1274: if ( length(L) == 1 ) {
! 1275: print(".",2);
! 1276: } else {
! 1277: print(" ",2); print(length(L),2); print(" ",2);
! 1278: }
! 1279: /* if (length(L)==1) {
! 1280: MP=append(MP,[NP[I]]);
! 1281: continue;
! 1282: } */
! 1283: R=append(R,L);
! 1284: }
! 1285: if ( PRINTLOG >= 3 )
! 1286: print("done");
! 1287: if ( length(R) )
! 1288: MP = idealsav(append(R,MP));
! 1289: LP = prime_irred(MP,VL);
! 1290: return LP;
! 1291: }
! 1292:
! 1293: /* radical decomposition program using by Zianni, Trager, Zacharias */
! 1294: /* R : factorized G-base in K[X], if FLAG then A is well comp. */
! 1295: def raddec(R,V,X,FLAG)
! 1296: {
! 1297: GR(A,R,V,irem(PRIMEORD,3));
! 1298: ZQ=zraddec(A,V); /* zero dimensional */
! 1299: /* if ( FLAG && length(ZQ) == 1 )
! 1300: return [R]; */
! 1301: if ( length(V) != length(X) )
! 1302: CQ=radcont(ZQ,V,X); /* contraction */
! 1303: else {
! 1304: for (CQ=[],I=length(ZQ)-1;I>=0;I--) {
! 1305: GR(R,ZQ[I],X,PRIMEORD);
! 1306: CQ=cons(R,CQ);
! 1307: }
! 1308: }
! 1309: return CQ;
! 1310: }
! 1311:
! 1312: /* radical decomposition for zero dimensional ideal */
! 1313: /* F is G-base w.r.t irem(PRIMEORD,3) */
! 1314: def zraddec(F,X)
! 1315: {
! 1316: if (length(F) == 1)
! 1317: return [F];
! 1318: Z=vvv;
! 1319: C=normposit(F,X,Z);
! 1320: /* C=[minimal polynomial, adding polynomial] */
! 1321: T=cdr(fctr(C[0]));
! 1322: if ( length(T) == 1 && T[0][1] == 1 )
! 1323: return [F];
! 1324: for (Q=[],I=0;I<length(T);I++) {
! 1325: if ( !C[1] ) {
! 1326: GR(P,cons(T[I][0],F),X,irem(PRIMEORD,3));
! 1327: } else {
! 1328: P=idealgrsub(append([T[I][0],C[1]],F),X,Z);
! 1329: }
! 1330: Q=cons(P,Q);
! 1331: }
! 1332: return Q;
! 1333: }
! 1334:
! 1335: /* contraction from V to X */
! 1336: def radcont(Q,V,X)
! 1337: {
! 1338: dp_ord(irem(PRIMEORD,3));
! 1339: for (R=[],I=length(Q)-1;I>=0;I--) {
! 1340: G=Q[I];
! 1341: for (E=1,J=0;J<length(G);J++)
! 1342: E = lcm(E,dp_hc(dp_ptod(G[J],V)));
! 1343: for (F=1,L=sqfr(E),K=1;K<length(L);K++)
! 1344: F *= L[K][0];
! 1345: H=localize(G,[F],X,PRIMEORD);
! 1346: R=cons(H,R);
! 1347: }
! 1348: return R;
! 1349: }
! 1350:
! 1351: /* A : polynomials (G-Base) */
! 1352: /* return [T,Y], T : minimal polynomial of Z, Y : append polynomial */
! 1353: def normposit(A,X,Z)
! 1354: {
! 1355: D = idim(A,X,irem(PRIMEORD,3)); /* dimension of the ideal Id(A) */
! 1356: for (I = length(A)-1;I>=0;I--) {
! 1357: for (J = length(X)-1; J>= 0; J--) {
! 1358: T=deg(A[I],X[J]);
! 1359: if ( T == D ) /* A[I] is a primitive polynomial of A */
! 1360: return [A[I],0];
! 1361: }
! 1362: }
! 1363: N=length(X);
! 1364: for (C = [],I = 0; I < N-1; I++)
! 1365: C=newvect(N-1);
! 1366: V=append(X,[Z]);
! 1367: ZD = (vars(A)==vars(X)); /* A is zero dim. over Q[X] or not */
! 1368: while( 1 ) {
! 1369: C = nextweight(C,cdr(X));
! 1370: for (Y=Z-X[0],I=0;I<N-1;I++) {
! 1371: Y-=C[I]*X[I+1]; /* new polynomial */
! 1372: }
! 1373: if ( !ZD ) {
! 1374: if ( version() == 940420 ) NOREDUCE = 1;
! 1375: else dp_gr_flags(["NoRA",1]);
! 1376: GR(G,cons(Y,A),V,3);
! 1377: if ( version() == 940420 ) NOREDUCE = 0;
! 1378: else dp_gr_flags(["NoRA",0]);
! 1379: for (T=G[length(G)-1],I = length(G)-2;I >= 0; I--)
! 1380: if ( deg(G[I],Z) > deg(T,Z) )
! 1381: T = G[I]; /* minimal polynomial */
! 1382: } else {
! 1383: T = minipoly(A,X,irem(PRIMEORD,3),Z-Y,Z);
! 1384: }
! 1385: if (deg(T,Z)==D)
! 1386: return [T,Y];
! 1387: }
! 1388: }
! 1389:
! 1390: def nextweight(C,V) /* degrevlex */
! 1391: {
! 1392: dp_ord(2);
! 1393: N = length(V);
! 1394: for (D=I=0;I<size(C)[0];I++)
! 1395: D += C[I];
! 1396: if ( C[N-1] == D ) {
! 1397: for (L=[],I=0;I<N-1;I++)
! 1398: L=cons(0,L);
! 1399: L = cons(D+1,L);
! 1400: return newvect(N,L);
! 1401: }
! 1402: CD=dp_vtoe(C);
! 1403: for (F=I=0;I<N;I++)
! 1404: F+=V[I];
! 1405: for (DF=dp_ptod(F^D,V);dp_ht(DF)!=CD;DF=dp_rest(DF));
! 1406: MD = dp_ht(dp_rest(DF));
! 1407: CC = dp_etov(MD);
! 1408: return CC;
! 1409: }
! 1410:
! 1411: def printsep(L)
! 1412: {
! 1413: for (I=0;I<length(L);I++) {
! 1414: if ( nmono(L[I][0]) == 1 )
! 1415: print(L[I][0],0);
! 1416: else {
! 1417: print("(",0); print(L[I][0],0); print(")",0);
! 1418: }
! 1419: if (L[I][1] > 1) {
! 1420: print("^",0); print(L[I][1],0);
! 1421: }
! 1422: if (I<length(L)-1)
! 1423: print("*",0);
! 1424: }
! 1425: }
! 1426:
! 1427: def idealgrsub(A,X,Z)
! 1428: {
! 1429: if ( PRIMEORD == 0 ) {
! 1430: dp_nelim(1); ORD = 8;
! 1431: } else
! 1432: ORD = [[2,1],[PRIMERORD,length(X)]];
! 1433: GR(G,A,cons(Z,X),ORD);
! 1434: for (R=[],I=length(G)-1;I>=0;I--) {
! 1435: V=vars(G[I]);
! 1436: for (J=0;J<length(V);J++)
! 1437: if (V[J]==Z)
! 1438: break;
! 1439: if (J==length(V))
! 1440: R=cons(G[I],R);
! 1441: }
! 1442: return R;
! 1443: }
! 1444:
! 1445: /* dimension of ideal */
! 1446: def idim(F,V,ORD)
! 1447: {
! 1448: return length(modbase(F,V,ORD));
! 1449: }
! 1450:
! 1451: def modbase(F,V,ORD)
! 1452: {
! 1453: dp_ord(ORD);
! 1454: for(G=[],I=length(F)-1;I>=0;I--)
! 1455: G = cons(dp_ptod(F[I],V),G);
! 1456: R = dp_modbase(G,length(V));
! 1457: for(S=[],I=length(R)-1;I>=0;I--)
! 1458: S = cons(dp_dtop(R[I],V),S);
! 1459: return S;
! 1460: }
! 1461:
! 1462: def dp_modbase(G,N)
! 1463: {
! 1464: E = newvect(N);
! 1465: R = []; I = 1;
! 1466: for (L = [];G != []; G = cdr(G))
! 1467: L = cons(dp_ht(car(G)),L);
! 1468: while ( I <= N ) {
! 1469: R = cons(dp_vtoe(E),R);
! 1470: E[0]++; I = 1;
! 1471: while( s_redble(dp_vtoe(E),L) ) {
! 1472: E[I-1] = 0;
! 1473: if (I < N)
! 1474: E[I]++;
! 1475: I++;
! 1476: }
! 1477: }
! 1478: return R;
! 1479: }
! 1480:
! 1481: def s_redble(M,L)
! 1482: {
! 1483: for (; L != []; L = cdr(L))
! 1484: if ( dp_redble(M,car(L)) )
! 1485: return 1;
! 1486: return 0;
! 1487: }
! 1488:
! 1489: /* FL : list of ideal Id(P,s) */
! 1490: def primedec_special(FL,VL)
! 1491: {
! 1492: if ( PRINTLOG ) {
! 1493: print("prime decomposition of the radical");
! 1494: }
! 1495: if ( PRINTLOG >= 2 )
! 1496: print("G-base factorization");
! 1497: PP = gr_fctr(FL,VL);
! 1498: if ( PRINTLOG >= 2 )
! 1499: print("special decomposition");
! 1500: SP = yokodec(PP,VL);
! 1501: for (L=[],I=length(SP)-1;I>=0;I--) {
! 1502: L=cons(idealnormal(SP[I]),L); /* head coef. > 0 */
! 1503: }
! 1504: SP = L;
! 1505: return SP;
! 1506: }
! 1507:
! 1508: /* PL : list of ideal Id(P,s) */
! 1509: def yokodec(PL,VL)
! 1510: {
! 1511: MSISTIME = 0; T = time()[0];
! 1512: for (R = [],I = 0; I<length(PL);I++) {
! 1513: PP = PL[I];
! 1514: if ( PRINTLOG >= 3 ) print(".",2);
! 1515: V = minalgdep(PP,VL,PRIMEORD);
! 1516: E = raddec(PP,V,VL,0);
! 1517: if ( length(E) >= 2 || !idealeq(E[0],PP) ) { /* prime check */
! 1518: T0 = time()[0];
! 1519: L=all_msis(PP,VL,PRIMEORD);
! 1520: MSISTIME += time()[0]-T0;
! 1521: E = yokodec_main(PP,L,VL);
! 1522: }
! 1523: R = append(R,E);
! 1524: }
! 1525: R = idealsav(R);
! 1526: R = prime_irred(R,VL);
! 1527: if ( PRINTLOG >= 3 ) {
! 1528: print("special dec time = ",0); print(time()[0]-T0);
! 1529: /* print(", ",0); print(MSISTIME); */
! 1530: }
! 1531: return R;
! 1532: }
! 1533:
! 1534: def yokodec_main(PP,L,VL)
! 1535: {
! 1536: AL = misset(L,VL); H = [1]; R = [];
! 1537: for (P=PP,I=0; I<length(AL); I++) {
! 1538: V = AL[I];
! 1539: if ( I && !ismad(P,AL[I],VL,PRIMEORD) )
! 1540: continue;
! 1541: U = raddec(PP,V,VL,0);
! 1542: R = append(R,U);
! 1543: for (I=0;I<length(U);I++)
! 1544: H = idealcap(H,U[I],VL,PRIMEORD);
! 1545: if ( idealeq(H,P) )
! 1546: break;
! 1547: F = wellsep(P,V,VL);
! 1548: GR(P,cons(F,P),VL,PRIMEORD);
! 1549: }
! 1550: return R;
! 1551: }
! 1552:
! 1553: /* output M.A.D. sets. AL : M.S.I.S sets */
! 1554: def misset(AL,VL)
! 1555: {
! 1556: for (L=[],D=0,I=length(AL)-1;I>=0;I--) {
! 1557: E = length(AL[I]);
! 1558: C = listminus(VL,AL[I]);
! 1559: if ( E == D )
! 1560: L = cons(C,L);
! 1561: else if ( E > D ) {
! 1562: L = [C]; D = E;
! 1563: }
! 1564: }
! 1565: return L;
! 1566: }
! 1567:
! 1568: end$
! 1569: