SINGULAR                                 /
 A Computer Algebra System for Polynomial Computations       /   version 3-1-6
                                                           0<
 by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann     \   Dec 2012
FB Mathematik der Universitaet, D-67653 Kaiserslautern        \
STDIN   1> //
STDIN   2. // version 2013
STDIN   3. //
STDIN   4. timer=1;
STDIN   5> //ring r=(integer,6),(u,v,a),(dp(2),lp);
STDIN   6. ring r=(integer,6),(u,v,a),dp;
STDIN   7> int b=0;
STDIN   8> matrix Z[5][5]; Z=Z+1;
STDIN   9> execute(read("fu_dataL.sing"));
STDIN  10> matrix tauL[1][5]; tauL=gen(1)*(1, b*v-a+u, 1, 1, a*u-b+v);
STDIN  11> matrix Z[315][315]; Z=Z+1;
// ** redefining Z **
STDIN  12> execute(read("fu_data0.sing"));
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STDIN  13> matrix t = matrix(rho(0),315,315);
STDIN  14> matrix t2=t*t;
STDIN  15> matrix t3=t2*t;
STDIN  16> module m=t3;
STDIN  17> matrix tau=m;
STDIN  18> vector vec=transpose(tau)[1];
STDIN  19> matrix tau[1][315]=vec;
// ** redefining tau **
STDIN  20> matrix mJ40 = matrix(J40,315,size(J40));
STDIN  21> 
STDIN  22. ideal IL; ideal I;
STDIN  23> IL=tauL*matrix(L0,5,size(L0)); IL=std(IL);
STDIN  24> 
STDIN  25. I=tau*matrix(J40,315,size(J40));
STDIN  26> I=std(I);
STDIN  27> 
STDIN  28. I=reduce(I,IL);
STDIN  29> I=I,IL; I=std(I);
STDIN  30> 
STDIN  31. ideal I1=I; ideal I2; int change=1; int i; int j; poly f; int count=0;
STDIN  32> while(change){change=0; count++;
STDIN  34.   for(i=1; i<=size(I1); i++){I2=0; 
STDIN  36.     for(j=1; j<=size(I1); j++){ if(j!=i){I2=I2,I1[j];}}
STDIN  38.     f=reduce(I1[i],I2);
STDIN  40.     if(f!=I1[i]){change=1; I1=I2,f; I1=std(I1); break;}
STDIN  42.   }
STDIN  44. }
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
// ** I2 is no standard basis
STDIN  45> I=I1;
STDIN  46> 
STDIN  47. matrix Z5[5][5]; Z5=Z5+1;
STDIN  48> 
STDIN  49. int i;
// ** redefining i **
STDIN  50> module mL=L0;
STDIN  51> for(i=1; i<=size(I); i++){mL=mL,module(I[i]*Z5);}
STDIN  52> mL=std(mL);
STDIN  53> 
STDIN  54. module m1L=mL;
STDIN  55> for( int i=0; i<9; i++ ){mL=mL,matrix(rhoL(i),5,5)*matrix(m1L,5,size(m1L));}
// ** redefining i **
STDIN  56> mL=groebner(mL);
STDIN  57> ideal I1L;
STDIN  58> I1L=tauL*matrix(mL,5,size(mL)); I1L=std(I1L);
STDIN  59> I1L=reduce(I1L,I);
STDIN  60> I1L;
I1L[1]=0
I1L[2]=0
I1L[3]=0
I1L[4]=0
I1L[5]=0
I1L[6]=0
I1L[7]=uv3a+5v3a2+3ua4+3u3v+v4+3va3+uv2+v2a+3v
I1L[8]=u2v2a+5v2a3+u4+uv3+4v3a+3ua3+3a4+5u2v+uva+5va2+2v2+5u+2a
I1L[9]=u3va+3v4a+v2a2+u3+5v3+3a3+3uv+2va+3
I1L[10]=u4a+uv3a+2v3a2+3ua4+3a5+4v4+5u2va+4uv2+2v2a+3ua
I1L[11]=0
I1L[12]=ua5+4uv4+2v4a+uv2a+2v2a2+3u3+3v3+5ua2+2a3+5uv+5va+5
I1L[13]=ua5+4uv4+2v4a+3uv2a+3u3+5v3+5ua2+2a3+uv+va+5
I1L[14]=v4a2+4uv3+3u3a+v3a+a4+3uva+va2+4u+3a
I1L[15]=v5a+5u5+5u2v3+5ua4+4a5+5v4+5u2va+3uv2+2v2a+u2+2ua+4a2+5v
I1L[16]=0
I1L[17]=0
I1L[18]=u6+v6+4v4a+3u2v2+3uv2a+4v2a2+5u3+5v3+ua2+2a3+3uv+3va+2
I1L[19]=0
STDIN  61> 
STDIN  62. I=I,I1L; I=std(I);
STDIN  63> 
STDIN  64. module m1L=mL;
// ** redefining m1L **
STDIN  65> module m2L=m1L;
STDIN  66> 
STDIN  67. for( i=0; i<9; i++ ){print(i);
STDIN  69.   mL=matrix(rhoL(i),5,5)*matrix(m1L,5,size(m1L));
STDIN  71.   mL=reduce(mL,m2L);
STDIN  73.   m2L=m2L,mL;
STDIN  75.   m2L=groebner(m2L);
STDIN  77.   print(size(m2L));
STDIN  79. }
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STDIN  80> 
STDIN  81. IL=tauL*matrix(m2L,5,size(m2L)); IL=reduce(IL,I); groebner(IL);
_[1]=0
STDIN  82> 
STDIN  83. module m3L=m2L;
STDIN  84> for( i=0; i<9; i++ ){print(i);
STDIN  86.   mL=matrix(rhoL(i),5,5)*matrix(m2L,5,size(m2L));
STDIN  88.   mL=reduce(mL,m3L);
STDIN  90.   m3L=m3L,mL;
STDIN  92.   m3L=groebner(m3L);
STDIN  94.   print(size(m3L));
STDIN  96. }
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STDIN  97> 
STDIN  98. reduce(m3L,m2L);
_[1]=0
_[2]=0
_[3]=0
_[4]=0
_[5]=0
_[6]=0
_[7]=0
_[8]=0
_[9]=0
_[10]=0
_[11]=0
_[12]=0
_[13]=0
_[14]=0
_[15]=0
_[16]=0
_[17]=0
_[18]=0
_[19]=0
_[20]=0
_[21]=0
_[22]=0
_[23]=0
_[24]=0
_[25]=0
_[26]=0
_[27]=0
_[28]=0
_[29]=0
_[30]=0
_[31]=0
_[32]=0
_[33]=0
_[34]=0
_[35]=0
_[36]=0
_[37]=0
_[38]=0
_[39]=0
_[40]=0
_[41]=0
_[42]=0
_[43]=0
_[44]=0
_[45]=0
_[46]=0
_[47]=0
_[48]=0
_[49]=0
_[50]=0
_[51]=0
_[52]=0
_[53]=0
_[54]=0
_[55]=0
_[56]=0
_[57]=0
_[58]=0
_[59]=0
_[60]=0
_[61]=0
_[62]=0
_[63]=0
_[64]=0
_[65]=0
_[66]=0
_[67]=0
_[68]=0
_[69]=0
_[70]=0
STDIN  99> IL=tauL*matrix(m2L,5,size(m2L)); IL=reduce(IL,I); groebner(IL);
_[1]=0
STDIN 100> 
STDIN 101. ideal I1;
// ** redefining I1 **
STDIN 102> m=matrix(rho(2),315,315)*matrix(J40,315,size(J40));
STDIN 103> I1=tau*matrix(m,315,size(m)); I1=reduce(I1,I);
STDIN 104> I=I,I1; I=std(I);
STDIN 105> print(size(I));
13
STDIN 106> I;
I[1]=2u2+2v
I[2]=2a3+4
I[3]=2uva+4va2+2v2+2u+2a
I[4]=2v3+4ua2+2a3+4uv+4va+4
I[5]=2uv2+4ua+2a2+2v
I[6]=a4+2a
I[7]=v2a2+ua2+a3+4uv+2va
I[8]=uva2+5va3+4v2a+4ua+a2
I[9]=u2a2+va2
I[10]=uv2a+3u3+3v3+5ua2+3uv+va+1
I[11]=u3a+3v3a+5uva+2va2+4v2+4u+a
I[12]=v5a+5u5+5u2v3+3v4+3u2va+uv2+u2+2a2+3v
I[13]=u6+3v6+5u2v2+3
STDIN 107> 
STDIN 108. module m1=J40;
STDIN 109> for( i=1; i<=size(I); i++){print(i);
STDIN 111.   m=module(I[i]*Z); m=std(m); m1=reduce(m1,m); print(size(m1));
STDIN 113.   m1=m,m1; m1=groebner(m1); print(size(m1));
STDIN 115. }
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STDIN 116> 
STDIN 117. module mm;
STDIN 118> module m2=m1; int j;
// ** redefining j **
STDIN 119> for( i=0; i<31; i++ ){print("--------");print(i);
STDIN 121.   m=matrix(rho(i),315,315)*matrix(J40,315,size(J40));
STDIN 123.   mm=t*m; m=m,mm; mm=t*m; m=m,mm;
STDIN 125.   m=reduce(m,m2); m2=m2,m; print(size(m));
STDIN 127.   if( size(m)>0 ){
STDIN 129.     I1=tau*matrix(m,315,size(m)); I1=reduce(I1,I);
STDIN 131.     I1=groebner(I1); print(I1);
STDIN 133.     if( size(I1)>1 | I1[1]!=0 ){
STDIN 135.       I=reduce(I,I1); I=I,I1; I=std(I); print("I=");print(I);print("----");
STDIN 137.       for( j=1; j<=size(I1); j++ ){m=module(I1[j]*Z); m2=m2,m;}
STDIN 139.     }
STDIN 141.   }
STDIN 143.   m2=groebner(m2);
STDIN 145.   print(size(m2));
STDIN 147. }
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STDIN 148> 
STDIN 149. module m3=m2; int j;
// ** redefining j **
STDIN 150> for( i=0; i<31; i++ ){print("--------");print(i);
STDIN 152.   m=matrix(rho(i),315,315)*matrix(m2,315,size(m2));
STDIN 154.   mm=t*m; m=m,mm; mm=t*m; m=m,mm;
STDIN 156.   m=reduce(m,m3); m3=m3,m; print(size(m));
STDIN 158.   if( size(m)>0 ){
STDIN 160.     I1=tau*matrix(m,315,size(m)); I1=reduce(I1,I);
STDIN 162.     I1=groebner(I1); print(I1);
STDIN 164.     if( size(I1)>1 | I1[1]!=0 ){
STDIN 166.       I=reduce(I,I1); I=I,I1; I=std(I); print("I=");print(I);print("----");
STDIN 168.       for( j=1; j<=size(I1); j++ ){m=module(I1[j]*Z); m3=m3,m;}
STDIN 170.     }
STDIN 172.   }
STDIN 174.   m3=groebner(m3);
STDIN 176.   print(size(m3));
STDIN 178. }
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STDIN 179> 
STDIN 180. module m4=m3; int j;
// ** redefining j **
STDIN 181> for( i=0; i<31; i++ ){print("--------");print(i);
STDIN 183.   m=matrix(rho(i),315,315)*matrix(m3,315,size(m3));
STDIN 185.   mm=t*m; m=m,mm; mm=t*m; m=m,mm;
STDIN 187.   m=reduce(m,m4); m4=m4,m; print(size(m));
STDIN 189.   if( size(m)>0 ){
STDIN 191.     I1=tau*matrix(m,315,size(m)); I1=reduce(I1,I);
STDIN 193.     I1=groebner(I1); print(I1);
STDIN 195.     if( size(I1)>1 | I1[1]!=0 ){
STDIN 197.       I=reduce(I,I1); I=I,I1; I=std(I); print("I=");print(I);print("----");
STDIN 199.       for( j=1; j<=size(I1); j++ ){m=module(I1[j]*Z); m4=m4,m;}
STDIN 201.     }
STDIN 203.   }
STDIN 205.   m4=groebner(m4);
STDIN 207.   print(size(m4));
STDIN 209. }
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STDIN 210> 
STDIN 211. m=reduce(m4,m3);
STDIN 212> groebner(m);
_[1]=0
STDIN 213> 
STDIN 214. I;
I[1]=2u2+2v
I[2]=2a3+4
I[3]=2uva+4va2+2v2+2u+2a
I[4]=2v3+4ua2+2a3+4uv+4va+4
I[5]=2uv2+4ua+2a2+2v
I[6]=a4+2a
I[7]=v2a2+ua2+a3+4uv+2va
I[8]=uva2+5va3+4v2a+4ua+a2
I[9]=u2a2+va2
I[10]=uv2a+3u3+3v3+5ua2+3uv+va+1
I[11]=u3a+3v3a+5uva+2va2+4v2+4u+a
I[12]=v5a+5u5+5u2v3+3v4+3u2va+uv2+u2+2a2+3v
I[13]=u6+3v6+5u2v2+3
STDIN 215> 
STDIN 216. // the gb for Q
STDIN 217. poly g(1)= a3+8;
STDIN 218> poly g(2)=2a3+1;
STDIN 219> poly g(3)=3a3+8;
STDIN 220> poly f(1)=g(1)*g(2)*g(3);
STDIN 221> poly f(2)=g(1)*g(3)*(u-a);
STDIN 222> poly f(3)=g(3)*(6u3-3a2u+a3+2);
STDIN 223> poly f(4)=336u4-792u3a+180u2a5+1272u2a2+846ua3+3264u-114a7-1405a4-3152a;
STDIN 224> poly f(5)=288v+336u3a2+216u2a3+2016u2-432ua4-2112ua-6a8+53a5+472a2;
STDIN 225> ideal Id; Id=f(1),f(2),f(3),f(4),f(5); Id=std(Id); Id;
Id[1]=2a3+4
Id[2]=a4+2a
STDIN 226> reduce(Id,I);
_[1]=0
_[2]=0
STDIN 227> reduce(I,Id);
_[1]=2u2+2v
_[2]=0
_[3]=2uva+4va2+2v2+2u+2a
_[4]=2v3+4ua2+4uv+4va
_[5]=2uv2+4ua+2a2+2v
_[6]=0
_[7]=v2a2+ua2+a3+4uv+2va
_[8]=uva2+5va3+4v2a+4ua+a2
_[9]=u2a2+va2
_[10]=uv2a+3u3+3v3+5ua2+3uv+va+1
_[11]=u3a+3v3a+5uva+2va2+4v2+4u+a
_[12]=v5a+5u5+5u2v3+3v4+3u2va+uv2+u2+2a2+3v
_[13]=u6+3v6+5u2v2+3
> Auf Wiedersehen.