Changeset 9a3ece in git
- Timestamp:
- May 26, 1999, 2:25:23 PM (24 years ago)
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- (u'spielwiese', '828514cf6e480e4bafc26df99217bf2a1ed1ef45')
- Children:
- 1598568cca6b50706674b0e35565a4c89bcfef83
- Parents:
- 9fd5f8a67b2159d3a039d7b0b34e704f6a617dfb
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- 1 edited
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Singular/LIB/normal.lib (modified) (48 diffs)
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Singular/LIB/normal.lib
r9fd5f8 r9a3ece 2 2 // normal.lib 3 3 // algorithms for computing the normalization based on 4 // the ideas of De Jong,Vasconcelos4 // the criterion of Grauert/Remmert and ideas of De Jong & Vasconcelos 5 5 // written by Gert-Martin Greuel and Gerhard Pfister 6 6 /////////////////////////////////////////////////////////////////////////////// 7 7 8 version="$Id: normal.lib,v 1.1 1 1999-03-15 12:07:19 SingularExp $";8 version="$Id: normal.lib,v 1.12 1999-05-26 12:25:23 obachman Exp $"; 9 9 info=" 10 LIBRARY: normal.lib : PROCEDURE FOR NORMALIZATION (I)11 12 normal(ideal I)13 // computes a set of rings such that their product is the14 // normalization of the reduced basering/I10 LIBRARY: normal.lib PROCEDURES FOR NORMALIZATION 11 12 normal(I); computes the normalization of basering/I 13 extraweight(R); intvec of variable weights of ring R 14 HomJJ(L); presentation of End_R(J) as affine ring, L a list 15 15 "; 16 16 … … 20 20 LIB "presolve.lib"; 21 21 /////////////////////////////////////////////////////////////////////////////// 22 22 static 23 23 proc isR_HomJR (list Li) 24 24 "USAGE: isR_HomJR (Li); Li = list: ideal SBid, ideal J, poly p … … 153 153 /////////////////////////////////////////////////////////////////////////////// 154 154 155 proc HomJJ (list Li ,list #)156 "USAGE: HomJJ (Li); Li = list: SBid,id,J,poly p155 proc HomJJ (list Li) 156 "USAGE: HomJJ (Li); Li = list: ideal SBid, ideal id, ideal J, poly p 157 157 ASSUME: R = P/id, P = basering, a polynomial ring, id an ideal of P, 158 158 SBid = standard basis of id, … … 166 166 endphi describes the canonical map R -> Hom_R(J,J) 167 167 _[2]: an integer which is 1 if phi is an isomorphism, 0 if not 168 NOTE: printlevel >=1: display comments (default: printlevel=0) 168 169 EXAMPLE: example HomJJ; shows an example 169 170 " … … 174 175 intvec rw,rw1; 175 176 list L; 176 if( size(#) >=1 ) 177 { 178 if( typeof(#[1]) == "int" ) { y = #[1]; } 179 } 177 178 y = printlevel-voice+2; // y=printlevel (default: y=0) 180 179 def P = basering; 181 180 ideal SBid, id, J = Li[1], Li[2], Li[3]; … … 219 218 //---------- computation of p*Hom(J,J) as R-ideal ----------------------------- 220 219 f = quotient(p*J,J); 221 if(y==1) 222 { 223 "the module Hom(rad(J),rad(J)) presented by the values on"; 224 "the non-zerodivisor"; 225 " "; 226 p; 227 " "; 228 f; 220 if ( y>=1 ) 221 { "// the module p*Hom(rad(J),rad(J)) = p*J:J, p a non-zerodivisor"; 222 223 "// p"; p; 224 "// f=p*J:J";f; 229 225 } 230 226 f2 = std(p); … … 271 267 L=lastRing; 272 268 L = insert(L,1,1); 273 dbprint( printlevel-voice+3,"// case R = Hom(J,J)");274 if(y ==1)269 dbprint(y,"// case R = Hom(J,J)"); 270 if(y>=1) 275 271 { 276 272 "R=Hom(rad(J),rad(J))"; … … 300 296 return(L); 301 297 } 302 if(y ==1)298 if(y>=1) 303 299 { 304 300 "R is not equal to Hom(rad(J),rad(J)), we have to try again"; … … 340 336 list L1; 341 337 342 //---------- computation of Hom(J,J) as ring ----------------------------------338 //---------- computation of Hom(J,J) as affine ring --------------------------- 343 339 // determine kernel of: R[T1,...,Tq] -> J:J >-> R[1/p]=R[t]/(t*p-1), 344 340 // Ti -> fi/p -> t*fi (p=f1=f[1]), to get ring structure. This is of course … … 350 346 T = matrix(ideal(T(1..q)),1,q); 351 347 Lin = ideal(T*syzf); 352 if(y ==1)348 if(y>=1) 353 349 { 354 350 "the ring structure of Hom(rad(J),rad(J)) as R-algebra"; … … 367 363 } 368 364 } 369 if(y ==1)365 if(y>=1) 370 366 { 371 367 "the quadratic relations"; … … 402 398 // export(endid); 403 399 // export(endphi); 404 if(y ==1)400 if(y>=1) 405 401 { 406 402 "the new ring after reduction of the number of variables"; … … 439 435 list Li = std(id),id,J,p; 440 436 list L = HomJJ(Li); 441 def end = L[1]; 442 setring end; 437 def end = L[1]; // defines ring L[1], containing ideals endid and endphi 438 setring end; // makes end the basering 443 439 end; 444 endid; 445 map psi = r,endphi; 440 endid; // end/endid is isomorphic to End(r/id) as ring 441 map psi = r,endphi;// defines the canonical map r/id -> End(r/id) 446 442 psi; 447 448 ring r = 32003,(x,y,z),dp;449 ideal id = x2-xy-xz+yz;450 ideal J =y-z,x-z;451 poly p = x+16001y+16001z;452 list Li = std(id),id,J,p;453 list L = HomJJ(Li,0);454 def end = L[1]; // defines ring L[1], containing ideals endid and endphi455 setring end; // makes end the basering456 end;457 endid; // end/endid is isomorphic to End(r/id) as ring458 459 443 } 460 444 461 445 /////////////////////////////////////////////////////////////////////////////// 462 446 proc normal(ideal id, list #) 463 "USAGE: normal(i,choose); i ideal,choose empty or 1 464 in case you choose 1 the factorizing Buchberger algorithm 465 is not used which is sometimes more efficient 466 RETURN: a list of rings L=R1,...,Rk, in each Ri are two ideals 467 Si,Mi such that the product of Ri/Mi is the normalization 468 Si is a standardbasis of Mi 469 NOTE: to use the rings: def r=L[i];setring r; 447 "USAGE: normal(i [,choose]); i ideal, choose empty or 1 448 if choose=1 the factorizing Buchberger algorithm is not used 449 (which is sometimes more efficient) 450 RETURN: a list L of rings, in each ring L[i] are two ideals 451 norid, normap such that the product of the L[i]/norid is the 452 normalization of basering/id and normap is the map from basering/id 453 to L[i]/norid 454 NOTE: to use the rings type: def r=L[i]; setring r; norid; normap; 455 increasing printlevel displays more comments (default: printlevel=0) 456 COMMENT: The algoritm works in principle for any basering and reduced ideal. 457 The implementation should work correctly for any global (p-) ordering, 458 however, this is not fully tested. 459 If the input ideal i is weighted homogeneous a weighted ordering may 460 be used (qhweight(i); computes weights). 470 461 EXAMPLE: example normal; shows an example 471 462 " … … 473 464 int i,j,y; 474 465 list result,prim,keepresult; 466 y=printlevel-voice+2; 467 468 if ( find(ordstr(basering),"s")+find(ordstr(basering),"M") != 0) 469 { 470 " 471 // Not implemented for this ordering, 472 // please change to global ordering!"; 473 return(result); 474 } 475 475 476 476 477 if(size(#)==0) … … 535 536 attrib(prim[1],"isRegInCodim2",0); 536 537 } 538 dbprint(y+1," 539 // If the name of your list is L type: 540 // def r=L[i]; setring r; norid; normap; 541 // then r/norid is the i-th ring of the normaliztion 542 // and normap the map of the basring to r/norid"); 543 537 544 return(normalizationPrimes(prim[1],maxideal(1))); 538 545 } … … 548 555 } 549 556 550 if(y ==1)557 if(y>=1) 551 558 { 552 559 "we have ";size(prim);"components"; … … 554 561 for(i=1;i<=size(prim);i++) 555 562 { 556 if(y ==1)563 if(y>=1) 557 564 { 558 565 "we are in loop";i; … … 592 599 } 593 600 } 601 602 dbprint(y+1," 603 // If the name of your list is L type: 604 // def r=L[i]; setring r; norid; normap; 605 // then r/norid is the i-th ring of the normaliztion 606 // and normap the map of the basring to r/norid"); 607 594 608 return(result); 595 609 } … … 597 611 example 598 612 { "EXAMPLE:"; echo = 2; 599 ring r = 0,(x,y,z),dp; 600 613 ring r=32003,(x,y,z),wp(2,1,2); 614 ideal i=z3-xy4; 615 list pr=normal(i); 616 pr; 617 def r1=pr[1]; 618 setring r1; 619 norid; 620 normap; 601 621 } 602 622 603 623 /////////////////////////////////////////////////////////////////////////////// 624 static 604 625 proc normalizationPrimes(ideal i,ideal ihp, list #) 605 626 "USAGE: normalizationPrimes(i); i prime ideal … … 608 629 S is a standardbasis of M 609 630 NOTE: to use the ring: def r=L[1];setring r; 631 printlevel >=1: display comments (default: printlevel=0) 610 632 EXAMPLE: example normalizationPrimes; shows an example 611 633 " 612 634 { 613 int y; 614 if(y==1) 635 int y = printlevel-voice+2; // y=printlevel (default: y=0) 636 637 if(y>=1) 615 638 { 616 639 "START one normalization loop with the ideal"; … … 625 648 def BAS=basering; 626 649 list result,keepresult1,keepresult2; 627 ideal J,SB,MB ,KK;650 ideal J,SB,MB; 628 651 int depth,lauf,prdim; 629 652 int ti=timer; … … 631 654 if(size(i)==0) 632 655 { 633 if(y ==1)656 if(y>=1) 634 657 { 635 658 "the ideal was the zero-ideal"; … … 637 660 execute "ring newR7="+charstr(basering)+",("+varstr(basering)+"),(" 638 661 +ordstr(basering)+");"; 639 ideal KK=ideal(0);640 ideal PP=fetch(BAS,ihp);641 export PP;642 export KK;662 ideal norid=ideal(0); 663 ideal normap=fetch(BAS,ihp); 664 export norid; 665 export normap; 643 666 result=newR7; 644 667 setring BAS; … … 647 670 } 648 671 649 if(y ==1)672 if(y>=1) 650 673 { 651 674 " "; … … 656 679 list SM=mstd(i); 657 680 658 if(y ==1)681 if(y>=1) 659 682 { 660 683 " the dimension is "; … … 727 750 if(dim(JM[1])==-1) 728 751 { 729 if(y ==1)752 if(y>=1) 730 753 { 731 754 "the ideal was smooth"; … … 736 759 def newR6=LL[1]; 737 760 setring newR6; 738 ideal KK=endid; 739 ideal PP=endphi; 740 // execute "ring newR6="+charstr(basering)+",("+varstr(basering)+"),(" 741 // +ordstr(basering)+");"; 742 // ideal KK=fetch(BAS,MB); 743 // ideal PP=fetch(BAS,ihp); 744 export PP; 745 export KK; 761 ideal norid=endid; 762 ideal normap=endphi; 763 export norid; 764 export normap; 746 765 result=newR6; 747 766 setring BAS; … … 753 772 if((dim(SM[1])==0)&&(homog(SM[2])==1)) 754 773 { 755 if(y ==1)774 if(y>=1) 756 775 { 757 776 "the ideal was zero-dimensional and homogeneous"; … … 762 781 def newR5=LL[1]; 763 782 setring newR5; 764 ideal KK=endid; 765 ideal PP=endphi; 766 // execute "ring newR5="+charstr(basering)+",("+varstr(basering)+"),(" 767 // +ordstr(basering)+");"; 768 // ideal KK=fetch(BAS,MB); 769 // ideal PP=fetch(BAS,ihp); 770 export PP; 771 export KK; 783 ideal norid=endid; 784 ideal normap=endphi; 785 export norid; 786 export normap; 772 787 result=newR5; 773 788 setring BAS; … … 781 796 &&(homog(SM[2])==1)) 782 797 { 783 if(y ==1)798 if(y>=1) 784 799 { 785 800 "the ideal defines a line"; … … 790 805 def newR4=LL[1]; 791 806 setring newR4; 792 ideal KK=endid; 793 ideal PP=endphi; 794 // execute "ring newR4="+charstr(basering)+",("+varstr(basering)+"),(" 795 // +ordstr(basering)+");"; 796 // ideal KK=fetch(BAS,MB); 797 // ideal PP=fetch(BAS,ihp); 798 export PP; 799 export KK; 807 ideal norid=endid; 808 ideal normap=endphi; 809 export norid; 810 export normap; 800 811 result=newR4; 801 812 setring BAS; … … 812 823 attrib(SM[2],"isCompleteIntersection",1); 813 824 attrib(SM[2],"isEquidimensional",1); 814 if(y ==1)825 if(y>=1) 815 826 { 816 827 "it is a complete Intersection"; 817 828 } 818 829 } 819 // if(attrib(i,"isIsolatedSingularity")==0)820 // {821 // list L=sres(SM[2],0);822 // prdim=ncols(betti(L))-1;823 // depth=nvars(basering)-prdim;824 // }825 826 // if(attrib(SM[2],"isCohenMacaulay")==0)827 // {828 // test for CohenMacaulay829 // if((dim(SM[1]))==depth)830 // {831 // attrib(SM[2],"isCohenMacaulay",1);832 // "it is CohenMacaulay";833 // }834 // }835 830 836 831 //compute the singular locus+lower dimensional components … … 841 836 J=minor(jacob(SM[2]),nvars(basering)-dim(SM[1])); 842 837 //ti=timer; 843 if(y ==1)838 if(y>=1) 844 839 { 845 840 "SB of singular locus will be computed"; … … 848 843 849 844 //kills the embeded components 850 // if(homog(SM[2])==1)851 // {852 // sin=sat(sin,maxideal(1))[1];853 // }854 845 855 846 list JM=mstd(sin); 856 if(y ==1)847 if(y>=1) 857 848 { 858 849 " "; … … 867 858 if(dim(JM[1])==-1) 868 859 { 869 if(y ==1)860 if(y>=1) 870 861 { 871 862 "it is smooth"; … … 874 865 intvec rw; 875 866 list LL=substpart(MB,ihp,0,rw); 876 // execute "ring newR3="+charstr(basering)+",("+varstr(basering)+"),("877 // +ordstr(basering)+");";878 867 def newR3=LL[1]; 879 868 setring newR3; 880 // ideal KK=fetch(BAS,MB); 881 // ideal PP=fetch(BAS,ihp); 882 ideal KK=endid; 883 ideal PP=endphi; 884 export PP; 885 export KK; 869 ideal norid=endid; 870 ideal normap=endphi; 871 export norid; 872 export normap; 886 873 result=newR3; 887 874 setring BAS; … … 916 903 if(size(qAnn)==0) 917 904 { 918 if(y ==1)905 if(y>=1) 919 906 { 920 907 " "; … … 928 915 //Hom(I,R)=R 929 916 list RR; 930 // list RR=SM[1],maxideal(1),SL[1];931 // "test with Hom(I,R)";932 // ti=timer;933 // if(isR_HomJR(RR)==1)934 // {935 // "it was normal";936 // timer-ti;937 // SB=SM[1];938 // SM=SM[2];939 // export SB,MB;940 // result=BAS;941 // return(result);942 // }943 // timer-ti;944 917 RR=SM[1],SM[2],maxideal(1),SL[1]; 945 918 ti=timer; 946 919 RR=HomJJ(RR,y); 947 // timer-ti;948 920 if(RR[2]==0) 949 921 { … … 961 933 execute "ring newR7="+charstr(basering)+",("+varstr(basering)+"),(" 962 934 +ordstr(basering)+");"; 963 ideal KK=fetch(BAS,MB);964 ideal PP=fetch(BAS,ihp);965 export PP;966 export KK;935 ideal norid=fetch(BAS,MB); 936 ideal normap=fetch(BAS,ihp); 937 export norid; 938 export normap; 967 939 result=newR7; 968 940 setring BAS; … … 1014 986 { 1015 987 list RR; 1016 // "test with Hom(I,I) before the radical computation";1017 //again test for normality1018 //Hom(I,R)=R1019 // list RR=SM[1],JM[2],SL[1];1020 // "test with Hom(I,R)";1021 // ti=timer;1022 // if(isR_HomJR(RR)==1)1023 // {1024 // "it was normal";1025 // timer-ti;1026 // SB=SM[1];1027 // SM=SM[2];1028 // export SB,MB;1029 // result=BAS;1030 // return(result);1031 // }1032 // timer-ti;1033 1034 // list RR=SM[1],SM[2],JM[2],SL[1];1035 // ti=timer;1036 988 list RS; 1037 // list RS=HomJJ(RR);1038 // timer-ti;1039 // if(RS[2]==0)1040 // {1041 // "Hom(I,I) was sucessfull without radical";1042 // def newR=RS[1];1043 // setring newR;1044 // list tluser=normalizationPrimes(SM);1045 // setring BAS;1046 // return(tluser);1047 // }1048 989 1049 990 //now we have to compute the radical 1050 if(y ==1)991 if(y>=1) 1051 992 { 1052 993 "radical computation"; 1053 994 } 1054 // ti=timer;1055 1056 995 1057 996 if((attrib(JM[2],"isRad")==0)&&(attrib(SM[2],"isEquidimensional")==0)) … … 1077 1016 } 1078 1017 } 1079 // timer-ti;1080 1018 1081 1019 JM=J,J; 1082 1020 1083 1021 //evtl. fuer SL[1] anderen Nichtnullteiler aus J waehlen 1084 // SL=simplify(reduce(J,SM[1]),2);1085 // Ann=quotient(SM[2],SL[1]);1086 // qAnn=simplify(reduce(Ann,SM[1]),2);1087 // if(size(qAnn)!=0)1088 // {1089 // keepresult1=normalizationPrimes(qAnn+SM[2]);1090 // keepresult2=normalizationPrimes(SL[1]+SM[2]);1091 // for(lauf=1;lauf<=size(keepresult2);j++)1092 // {1093 // keepresult1=insert(keepresult1,keepresult2[lauf]);1094 // }1095 // return(keepresult1);1096 // }1097 1022 RR=SM[1],SM[2],JM[2],SL[1]; 1098 1023 1099 1024 // evtl eine geeignete Potenz von JM? 1100 if(y ==1)1025 if(y>=1) 1101 1026 { 1102 1027 "compute Hom(rad(J),rad(J)):"; 1103 1028 } 1104 // ti=timer;1105 1029 1106 1030 RS=HomJJ(RR,y); 1107 // timer-ti;1108 1031 1109 1032 if(RS[2]==1) … … 1112 1035 setring lastR; 1113 1036 map psi1=BAS,endphi; 1114 ideal KK=endid;1115 ideal PP=psi1(ihp);1116 export PP;1117 export KK;1037 ideal norid=endid; 1038 ideal normap=psi1(ihp); 1039 export norid; 1040 export normap; 1118 1041 setring BAS; 1119 // return(RS[1]);1120 1042 return(lastR); 1121 1043 } … … 1129 1051 list tluser= 1130 1052 normalizationPrimes(endid,psi(ihp),simplify(psi(MJ)+endid,4)); 1131 // normalizationPrimes(endid);1132 1053 setring BAS; 1133 1054 return(tluser); … … 1139 1060 execute "ring newR1="+charstr(basering)+",("+varstr(basering)+"),(" 1140 1061 +ordstr(basering)+");"; 1141 if(y ==1)1062 if(y>=1) 1142 1063 { 1143 1064 "zero-divisor found"; … … 1164 1085 ideal new2=quotient(SM[2],Ann)+SM[2]; 1165 1086 // evtl. qAnn nehmen 1166 // ideal new2=SL[1]+SM[2];1167 1087 execute "ring newR2="+charstr(basering)+",("+varstr(basering)+"),(" 1168 1088 +ordstr(basering)+");"; … … 1196 1116 example 1197 1117 { "EXAMPLE:";echo = 2; 1198 LIB"normal.lib";1199 1118 //Huneke 1200 ring qr=31991,(a,b,c,d,e),dp; 1201 ideal i= 1202 5abcde-a5-b5-c5-d5-e5, 1203 ab3c+bc3d+a3be+cd3e+ade3, 1204 a2bc2+b2cd2+a2d2e+ab2e2+c2de2, 1205 abc5-b4c2d-2a2b2cde+ac3d2e-a4de2+bcd2e3+abe5, 1206 ab2c4-b5cd-a2b3de+2abc2d2e+ad4e2-a2bce3-cde5, 1207 a3b2cd-bc2d4+ab2c3e-b5de-d6e+3abcd2e2-a2be4-de6, 1208 a4b2c-abc2d3-ab5e-b3c2de-ad5e+2a2bcde2+cd2e4, 1209 b6c+bc6+a2b4e-3ab2c2de+c4d2e-a3cde2-abd3e2+bce5; 1210 1211 list pr=normal(i); 1212 def r1=pr[1]; 1213 setring r1; 1214 KK; 1119 ring qr=31991,(a,b,c,d,e),dp; 1120 ideal i= 1121 5abcde-a5-b5-c5-d5-e5, 1122 ab3c+bc3d+a3be+cd3e+ade3, 1123 a2bc2+b2cd2+a2d2e+ab2e2+c2de2, 1124 abc5-b4c2d-2a2b2cde+ac3d2e-a4de2+bcd2e3+abe5, 1125 ab2c4-b5cd-a2b3de+2abc2d2e+ad4e2-a2bce3-cde5, 1126 a3b2cd-bc2d4+ab2c3e-b5de-d6e+3abcd2e2-a2be4-de6, 1127 a4b2c-abc2d3-ab5e-b3c2de-ad5e+2a2bcde2+cd2e4, 1128 b6c+bc6+a2b4e-3ab2c2de+c4d2e-a3cde2-abd3e2+bce5; 1129 1130 list pr=normalizationPrimes(i); 1131 def r1=pr[1]; 1132 setring r1; 1133 norid; 1134 normap; 1215 1135 } 1216 1217 proc substpart(ideal endid,ideal endphi, int homo, intvec rw) 1136 /////////////////////////////////////////////////////////////////////////////// 1137 static 1138 proc substpart(ideal endid, ideal endphi, int homo, intvec rw) 1218 1139 { 1219 1140 def newRing=basering; … … 1297 1218 return(L); 1298 1219 } 1299 1220 /////////////////////////////////////////////////////////////////////////////// 1221 static 1300 1222 proc diagon(ideal i) 1301 1223 { … … 1308 1230 ///////////////////////////////////////////////////////////////////////////// 1309 1231 /* 1232 Examples: 1310 1233 LIB"normal.lib"; 1311 int aa=timer;list pr=normal(i);timer-aa;1312 1313 1314 1315 1234 //Huneke 1316 1235 ring qr=31991,(a,b,c,d,e),dp;
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