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Annotation of OpenXM/src/k097/lib/minimal/minimal-note-ja.txt, Revision 1.6

1.6     ! takayama    1: $OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.5 2000/06/14 07:44:04 takayama Exp $
1.1       takayama    2:
                      3: SpairAndReduction() :
                      4:    $BM?$($i$l$?(B pair $B$r(B reduction $B$9$k(B.
                      5:    V-minimal $B$KI,MW$+$I$&$+$NH=Dj$b$9$k(B.
                      6:
                      7: SpairAndReduction2():
                      8:   tower2 = StowerOf(tower,level-1);
                      9:   SsetTower(tower2);
                     10:   /** sm1(" show_ring ");   */
                     11:
                     12:   $BM?$($i$l$?(B pair $B$r(B reduction $B$9$k$?$a$N(B schreyer order
                     13:   $B$r@_Dj$9$k(B.  Resolution $B$N?<$5$K1~$8$F(B, tower $B$b?<$/$9$kI,MW$,$"$k(B.
                     14:
                     15:
                     16:   if (IsConstant(t_syz[i])){
                     17:
                     18:   Syzygy $B$r$_$F(B, $BDj?t@.J,$,$J$$$+(B check.
                     19:   t_syz[i] $B$,Dj?t@.J,$G$"$l$P(B, $B0l$DA0$N(B GB $B$N9=@.MWAG$G$"$k(B
                     20:   g_i $B$,M>J,$J(B GB $B$G$"$k2DG=@-$,$?$+$$(B.
                     21:   SpairAndReduction() ( LaScala-Stillman $B$NJ}K!(B) $B$H$N@09g@-$r$H$k$?$a(B
                     22:   g_i $B$r(B tmp[0] $B$KBeF~$7(B ( reduction $B$G$-$J$+$C$?$U$j$r$9$k(B )
                     23:   g_i $B$N(B V-degree $B$r$7$i$Y$k(B.
                     24:
                     25:
                     26: Sannfs2_laScala2()
                     27: Sannfs3_laScala2()  $B$r:n$k(B.
                     28:
                     29: $BFs$D$N%"%k%4%j%:%`$NHf3S(B.
                     30: In(11)=sm1_pmat(a1[1]); $B$N=gHV$r$+$($k(B.
                     31:  [
                     32:    [    3*Dx^2*h , 0 , Dy , -Dz ]
                     33:    [    6*x*Dy*Dz^2-9*y*Dx^2*h , -2*x*Dy*Dz^2+3*y*Dx^2*h , -2*x*Dx-3*y*Dy , 0]
                     34:    [    2*x*Dx+3*z*Dz-h^2 , y*Dy-z*Dz , 0 , 0 ]
                     35:    [    2*x*Dy*Dz , 0 , z , -y ]
                     36:
                     37:    [    0 , 2*x*Dy^2*Dz-3*z*Dx^2*h , 0 , 2*x*Dx+3*z*Dz ]
                     38:  ]
                     39: In(12)=sm1_pmat(a2[1]);
                     40:  [
                     41:    [    3*Dx^2*h , 0 , Dy , -Dz ]
                     42:    [    6*x*Dy*Dz^2-9*y*Dx^2*h , -2*x*Dy*Dz^2+3*y*Dx^2*h , -2*x*Dx-3*y*Dy, 0 ]
                     43:    [    2*x*Dx+3*z*Dz-h^2 , y*Dy-z*Dz , 0 , 0 ]
                     44:    [    2*x*Dy*Dz , 0 , z , -y ]
                     45:
                     46:    [    9*z*Dx^2*h , 2*x*Dy^2*Dz-3*z*Dx^2*h , 3*z*Dy , 2*x*Dx ]
                     47:    [    2*x*Dx*Dz^2+3*z*Dz^3+5*Dz^2*h^2 , y*Dy*Dz^2-z*Dz^3-2*Dz^2*h^2 , 0 , 0 ]
                     48:  ]
                     49: In(13)=
                     50:
                     51: ----------------------
                     52: In(16)=sm1_pmat(a1[2]);
                     53:  [
                     54:    [    -2*x*Dx-3*y*Dy-3*z*Dz-6*h^2 , -Dy , -Dz , 3*Dx^2*h , 3*Dy^2 , 3*Dy*Dz , -2*x*Dy , 2*x*Dz , 0 ]
                     55:    [    3*y*z , z , y , -2*x*Dy*Dz , -3*z*Dy , 2*x*Dx , 2*x*z , -2*x*y , 0 ]
                     56:  ]
                     57: In(17)=sm1_pmat(a2[2]);
                     58:  [
                     59:    [    -y , 2*x*Dy*Dz , z , 0 , 2*x*Dx , 0 ]
                     60:    [    -Dz , 3*Dx^2*h , Dy , -2*x*Dx-3*y*Dy-3*h^2 , -3*Dy*Dz , 0 ]
                     61:  ]
                     62: In(18)=
                     63:
1.2       takayama   64: ---------------------------
                     65:
                     66: May 22, (Tue),  5:50 (Spain local time, 12:50 JST)
                     67:
                     68: kan96xx/Kan/resol.c $B$G(B,
                     69:    RemoveRedundantInSchreyerSkelton = 0
                     70: $B$KJQ$($F(B ($B$3$N(B option $B$b$"$?$i$7$/2C$($k(B), schreyer $B$,@5$7$/F0$/$+(B
                     71: $BD4$Y$k$3$H$K$9$k(B.
                     72: ( commit $B$O(B kan96xx $B$H(B k097 $BN>J}$9$Y$7(B.)
                     73:
                     74: test8() $B$G(B sm1 $B$G=q$$$?J}$N(B Schreyer $B$r8+$k$H(B,
                     75:    RemoveRedundantInSchreyerSkelton = 1
                     76: $B$G$b(B,
                     77: kernel = image
                     78: $B$H$J$C$F$$$k$N$G0J8e$3$N(B option $B$O(B 1 $B$N$^$^;H$&$3$H$H$9$k(B.
                     79: $BMW$9$k$K(B k0 $B$N%3!<%I$,$I$&$d$i$*$+$7$$$i$7$$(B.
1.4       takayama   80: ==>
                     81: 6/8 $B$N%N!<%H$h$j(B.
                     82: syzygy $B$r(B homogenization $B$r2p$7$F7W;;$9$k$N$OLdBj$"$j(B.
                     83: --> usage of isExact
                     84: $BMW$9$k$K(B kernel = image $B$N%3!<%I$bJQ(B.  Homogenized $B$N$^$^$d$kI,MW$"$j(B.
1.3       takayama   85:
                     86: -----------------------------------
                     87: June 8, 2000 (Thu), 9:10 (Spain local time)
                     88: hol.sm1 :  gb_h, syz_h, isSameIdeal, isSameIdeal_h
                     89: complex.sm1 :  isExact, isExact_h
                     90:
                     91: syzygy $B$r(B homogenization $B$r2p$7$F7W;;$9$k$N$OLdBj$"$j(B.
                     92: --> usage of isExact
                     93:
                     94: [(Homogenize_vec) 0] system_variable : vector $B$N(B homogenize $B$r$7$J$$(B.
                     95: (grade) (module1v) switch_function : vector $BJQ?t$O(B, total
                     96:        degree $B$K?t$($J$$(B.
                     97: ==> $BL58B%k!<%W$KCm0U(B   ---> gb_h, syz_h  $B$N(B usage.
                     98:
                     99: minimal-test.k $B$N(B ann(x^3-y^2*z^2) $B$N(B laplace $BJQ49$N(B
                    100: betti $B?t$,JQ(B, exact $B$G$J$$(B, $B$r(B isExact_h $B$G(B check
                    101: $B$7$h$&(B.
                    102:
                    103: minimal-test.k
                    104: test10();
                    105:   LaScala-Stillman $B$NJ}K!$G$D$/$C$?(B, schreyer resol $B$,(B exact $B$+(B
                    106:   $BD4$Y$k(B.
                    107:   $BNcBj$O(B, ann(1/(x^3-y^2 z^2)) $B$N(B Laplace $BJQ49(B.
1.4       takayama  108:   ==> OK.  IsExact_h $B$G$7$i$Y$k(B.  (IsExact $B$O$@$a$h(B)
                    109:
                    110: June 8, 2000 (Thu), 19:35
                    111: load["minimal-test.k"];;
                    112: test11();
                    113:   LaScala-Stillman $B$NJ}K!$G$D$/$C$?(B, minimal resol $B$,(B exact $B$+(B
                    114:   $BD4$Y$k(B.
                    115:   $BNcBj$O(B, ann(1/(x^3-y^2 z^2)) $B$N(B Laplace $BJQ49(B.
                    116:
                    117: SwhereInTower $B$r;H$&$H$-$O(B,
                    118: SsetTower() $B$G(B gbList $B$rJQ99$7$J$$$H$$$1$J$$(B.
                    119: $B$b$A$m$s;HMQ$7$?$i(B, $B$=$l$rLa$9$3$H(B.
                    120: SpairAndReduction, SpairAndReduction2 $B$G(B,
                    121:   SsetTower(StowerOf(tower,level));
                    122:   pos = SwhereInTower(syzHead,tower[level]);
                    123:
                    124:   SsetTower(StowerOf(tower,level-1));
                    125:   pos2 = SwhereInTower(tmp[0],tower[level-1]);
                    126: $B$H(B, SwhereInTower $B$NA0$K(B setTower $B$r$/$o$($?(B.
                    127: ( $B0c$&%l%Y%k$G$NHf3S$N$?$a(B.)
                    128:
                    129: IsExact_h $B$O(B, 0 $B%Y%/%H%k$r4^$`>l9g(B, $B$?$@$7$/F0:n$7$J$$$h$&$@(B.
                    130: test11().
                    131: test11a() $B$G(B, 0 $B%Y%/%H%k$r<j$G=|$$$?9TNs$N(B exactness $B$r%A%'%C%/(B. ==> OK.
                    132:
                    133:
                    134: ---------------------------------
                    135: June 9, 6:20
                    136: SpairAndReduction
                    137: $B$H(B
                    138: SpairAndReduction2
                    139: $B$N0c$$(B.
                    140: SpairAndReduction  :  SlaScala  (LaScala-Stillman's algorithm $B$G;H$&(B)
                    141: SpairAndReduction2 :  Sschreyer (schreyer  algorithm $B$G;H$&(B, laScala $B$O$J$7(B.)
                    142:
                    143: 0 $B$r<+F0$G=|$/%3!<%I$r=q$3$&(B.
                    144:
                    145: SpruneZeroRow() $B$r(B Sminimal() $B$K2C$($?(B.
                    146: test11() $B$b@5$7$/F0:n$9$k$O$:(B.
                    147: IsExact_h $B$O(B schreyer $B$r(B off $B$7$F(B, ReParse $B$7$F$+$i(B,
                    148: $B8F$S=P$9$3$H(B.
                    149:
                    150:
                    151: #ifdef TOTAL_STRATEGY
                    152:   return(StotalDegree(f)+Sdegree(tower[level-2,i],tower,level-1));
                    153: #endif
                    154:   /* Strategy must be compatible with ordering.  */
                    155:   /* Weight vector must be non-negative, too.  */
                    156:   /* See Sdegree, SgenerateTable, reductionTable. */
                    157:   wd = Sord_w(f,ww);
                    158:   return(wd+Sdegree(tower[level-2,i],tower,level-1));
                    159: TOTAL_STRATEGY $B$rMQ$$$kI,MW$,$"$k$N$G$O(B??
                    160: Example 1:  Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
                    161:           v=[[2*x*Dx + 3*y*Dy+6, 0],
                    162:              [3*x^2*Dy + 2*y*Dx, 0],
                    163:              [0,  x^2+y^2],
                    164:              [0,  x*y]];
                    165:          a=Sminimal(v);
                    166: strategy $B$,$*$+$7$$$H$$$C$F$H$^$k(B. $BM}M3$O(B?
                    167:
                    168: a=test_ann3("x^3+y^3+z^3); $B$O;~4V$,$+$+$j$=$&(B.
                    169: a=test_ann3("x^3+y^3"); OK.
                    170: a=test_ann3("x^2+y^2+z"); OK.
                    171:
                    172:
                    173: $B>e$N(B example 1 $B$N%(%i!<(B $B$N8+J}(B:
                    174: Processing [    1 , 3 ]    Strategy = 2
                    175:      1 $B$N(B 3 $BHVL\$N(B spair $B$N(B reduction $B$r=hM}Cf(B.
                    176:      In(7)=reductionTable:
                    177:     [[ 1 , 1 , 1 , 2 , 2 , 3 ]  , [    3 , 2 , 1 , 2 , 3 ]  , [    2 ]  ]
                    178:                                                    -- $B$3$l(B.
                    179: SpairAndReduction:
                    180: [    p and bases  , [    [    0 , 3 ]  , [    y*h , -x ]  ]  , [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , %[null] ]  ]
                    181: 0 $B$N(B 0 $BHVL\$H(B 3 $BHVL\(B $B$N(B spair $B$r7W;;$7$F(B, 0 $B%l%Y%k$N(B gb $B$G(B reduction.
                    182: [ 1 , 1 , 1 , 2 , 2 , 3 ] $B$K$"$k$h$&$K(B, strategy 3 $B0J30$O7W;;$:$_(B.
                    183: ( $B7W;;$7$F$J$$$b$N$O(B %[null] $B$H$J$C$F$k(B. )
                    184: [    level= , 1 ]
                    185: [    tower2= , [    [   ]  ]  ]   ( $B0lHV2<$J$N$G(B, tower $B$O$J$7$h(B. )
                    186: [    y*h , -es^3*x ]
                    187: [gi, gj] = [    2*x*Dx+3*y*Dy+6*h^2 , 2*y*Dx*h+3*x^2*Dy ]
                    188: 1
                    189: Reduce the element 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy
                    190: by  [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , %[null] ]
                    191: result is [    3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , 1 , [    0 , 0 , 0 , 0 , 0 , 0 ]  ]
                    192: vdegree of the original = -1
                    193: vdegree of the remainder = -1
                    194: [    3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , [    y*h , 0 , 0 , -x , 0 , 0 ]  , 3 , 5 , -1 , -1 ]
                    195:
                    196: In(11)=freeRes:
                    197: [    [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy ]  , [    %[null] , [    0 , 0 , y^2 , 0 , -x , 0 ]  , [    0 , -y , x , 0 , 1 , 0 ]  , [    -y*h , 0 , 0 , x , 0 , 1 ]  , %[null] ]  , [    %[null] ]  ]
                    198: $B$r$_$l$P$o$+$k$h$&$K(B, SlaScala $B$G(B, freeRes $B$K$3$N85$,(B [0,5] $B$K2C$((B
                    199: $B$i$l$?(B.
                    200:
                    201: $B<!$K(B SnextI $B$,(B SlaScala $B$h$j8F$P$l$F$3$N%(%i!<(B.
                    202:         i = SnextI(reductionTable_tmp,strategy,redundantTable,
                    203:                    skel,level,freeRes);
                    204: In(22)=reductionTable:
                    205: [    [    1 , 1 , 1 , 2 , 2 , 3 ]  , [    3 , 2 , 1 , 2 , 3 ]  , [    2 ]  ]
                    206: $B$J$N$G(B, $B:G8e(B $B$N(B 2 $B$,=hM}$5$l$k$O$:$@$,(B,
                    207: In(25)=skel[2]:
                    208: [    [    [    0 , 2 ]  , [    1 , -y^2 ]  ]  ]
                    209: $B$N$h$&$K(B, 0 $BHVL\$H(B, 2 $BHVL\$N(B spair.
                    210: $B$7$+$7(B,
                    211: In(26)=bases:
                    212: [    %[null] , [    0 , 0 , y^2 , 0 , -x , 0 ]  , [    0 , -y , x , 0 , 1 , 0 ]  , [    -y*h , 0 , 0 , x , 0 , 1 ]  , %[null] ]
                    213: $B$N$h$&$K(B, 0 $BHVL\$O(B strategy 3 $B$J$N$G(B, $B$^$@$b$H$^$C$F$$$J$$(B.
                    214:
                    215: reductionTable_tmp=[    2 ]
                    216: See also reductionTable, strategy, level,i
                    217: ERROR(sm): error operator : SnextI: bases[i] or bases[j] is null for all combinations.
                    218: --- Engine error or interrupt : In function : Error of class PrimitiveObject
                    219:
                    220: Type in Cleards() to exit the debug mode and Where() to see the stack trace.
                    221: In(7)=reductionTable:
                    222: [    [    1 , 1 , 1 , 2 , 2 , 3 ]  , [    3 , 2 , 1 , 2 , 3 ]  , [    2 ]  ]
                    223: In(8)=strategy:
                    224: 2
                    225: In(9)=level:
                    226: 2
                    227:
                    228:    RemoveRedundantInSchreyerSkelton = 0
                    229: $B$H$7$F$bF1$8%(%i!<(B.
                    230:
                    231: -------------------------------------------------
                    232: test_ann3("x*y+y*z+z*x");    OK.
                    233:
                    234: 6/9 (Fri)
                    235: Sminimal $B$N<BAu$KAjJQ$o$i$:6lO+$7$F$^$9(B.
                    236: Sevilla $B$G$$$m$$$m$HD>$7$?7k2L(B,
                    237: Sminimal $B$O$&$^$/$&$4$1$P@5$7$$Ez$($r$@$7$F$k$_$?$$$G$9$,(B
                    238: (D<h> : homogenized Weyl $B$G(B ker = im $B$r(B check $B$7$F$k(B,
                    239:  V-adapted (strict) $B$+$I$&$+$N(B check routing $B$O$^$@=q$$$F$J$$(B),
                    240: strategy $B$,$&$^$/$&$4$+$J$/$F$H$^$k>l9g$b$"$j$^$9(B
                    241: ( strategy = 2 $B$N(B sp $B$r7W;;$9$k$N$K(B, strategy 3 $B$N(B $B85$rI,MW$H(B
                    242:   $B$7$?$j$9$k>l9g$"$j(B).
                    243:
                    244:
                    245: strategy $B$O(B
                    246: def Sdegree(f,tower,level) {
                    247:   local i,ww, wd;
                    248:   /* extern WeightOfSweyl; */
                    249:   ww = WeightOfSweyl;
                    250:   f = Init(f);
                    251:   if (level <= 1) return(StotalDegree(f));
                    252:   i = Degree(f,es);
                    253:   return(StotalDegree(f)+Sdegree(tower[level-2,i],tower,level-1));
                    254: }
                    255: $B$rMQ$$$F(B,
                    256:       ans_at_each_floor[j] = Sdegree(tower[i,j],tower,i+1)-(i+1)
                    257: $B$G7W;;$7$F$^$9(B.
                    258:
                    259: $B$$$/$D$+=PNO$r$D$1$F$*$-$^$9$N$G(B, $B8!F$(B!!!
                    260:
                    261: $BNc(B 1:
                    262: load["minimal-test.k"];;
                    263: a=test_ann3("x^3-y^2*z^2"); $B0z?t$N(B annihilating ideal $B$N(B laplace $BJQ49$N(B
                    264:                             homogenization $B$N(B resolution.
                    265:       weight vector $B$O(B (-1,-1,-1,1,1,1)
                    266:
                    267: In(4)=sm1_pmat(a[1]);
                    268:  [
                    269:   [   0 $B<!(B
                    270:     [    y*Dy-z*Dz ]
                    271:     [    -2*x*Dx-3*z*Dz+h^2 ]
                    272:     [    2*x*Dy*Dz^2-3*y*Dx^2*h ]
                    273:     [    2*x*Dy^2*Dz-3*z*Dx^2*h ]
                    274:   ]
                    275:   [   1 $B<!(B
                    276:     [    3*Dx^2*h , 0 , Dy , -Dz ]
                    277:     [    6*x*Dy*Dz^2-9*y*Dx^2*h , -2*x*Dy*Dz^2+3*y*Dx^2*h , -2*x*Dx-3*y*Dy , 0 ]
                    278:     [    0 , 2*x*Dy^2*Dz-3*z*Dx^2*h , 0 , 2*x*Dx+3*z*Dz ]
                    279:     [    2*x*Dx+3*z*Dz-h^2 , y*Dy-z*Dz , 0 , 0 ]
                    280:     [    2*x*Dy*Dz , 0 , z , -y ]
                    281:   ]
                    282:   [  2 $B<!(B
                    283:     [    -2*x*Dx-3*y*Dy-3*z*Dz-6*h^2 , -Dy , -Dz , 3*Dx^2*h , 3*Dy*Dz ]
                    284:     [    3*y*z , z , y , -2*x*Dy*Dz , 2*x*Dx ]
                    285:   ]
                    286:  ]
                    287: In(5)=
                    288:
                    289: $BNc(B 2:
                    290: load["minimal-test.k"];;
                    291: a=test_ann3("x*y+y*z+z*x");
                    292: In(6)=sm1_pmat(a[1]);
                    293:  [
                    294:   [  0 $B<!(B
                    295:     [    2*x*Dx+x*Dz-y*Dz+z*Dz+h^2 ]
                    296:     [    -2*y*Dy+x*Dz-y*Dz-z*Dz-h^2 ]
                    297:     [    -2*x*Dy+2*z*Dy+x*Dz-y*Dz+3*z*Dz+h^2 ]
                    298:     [    -2*y*Dx+2*z*Dx-x*Dz+y*Dz+3*z*Dz+h^2 ]
                    299:   ]
                    300:   [  1 $B<!(B
                    301:     [    y-z , x-z , -y , x ]
                    302:     [    2*Dy-2*Dz , 2*Dx-2*Dz , 2*Dx+2*Dz , -2*Dy-2*Dz ]
                    303:     [    2*y*Dx-2*z*Dx+x*Dz-y*Dz-3*z*Dz-2*h^2 , 0 , 0 , 2*x*Dx+x*Dz-y*Dz+z*Dz+2*h^2 ]
                    304:     [    2*y*Dy-2*z*Dy+y*Dz-z*Dz+h^2 , 2*x*Dz-y*Dz+2*z*Dz+h^2 , -x*Dz+z*Dz , 2*x*Dy+x*Dz ]
                    305:     [    -2*y*Dy+2*z*Dy+y*Dz-z*Dz , y*Dz-4*z*Dz , -2*y*Dx+2*z*Dx-y*Dz+2*z*Dz , -2*z*Dy+y*Dz-3*z*Dz ]
                    306:   ]
                    307:   [  2 $B<!(B
                    308:     [    -2*y*Dx+2*z*Dx-y*Dz+2*z*Dz , x*y-x*z-y*z+z^2 , y-z , y , x+y-z ]
                    309:     [    -6*Dx*Dz-2*Dz^2 , x*Dz+y*Dz-5*z*Dz-4*h^2 , -2*Dy+2*Dz , 2*Dx+2*Dz , 4*Dz ]
                    310:   ]
                    311:  ]
                    312: In(7)=
                    313:
                    314: $BNc(B 3:  $B$&$^$/9T$+$J$$Nc(B:
                    315:
                    316: Example 1:  Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
                    317:           v=[[2*x*Dx + 3*y*Dy+6, 0],
                    318:              [3*x^2*Dy + 2*y*Dx, 0],
                    319:              [0,  x^2+y^2],
                    320:              [0,  x*y]];
                    321:          a=Sminimal(v);
                    322: strategy $B$,$*$+$7$$$H$$$C$F$H$^$k(B. $BM}M3$O(B?
                    323: Negative weight vector $B$r;H$o$J$$$H$-$A$s$HF0$-$^$9(B.
                    324:
                    325:
                    326: DEBUG $B=PNO(B:
                    327: rf= [
                    328:   [
                    329:    [   Schreyer frame.
                    330:      [    0 , y^3 , 0 , 0 , -x^2 , 0 ]
                    331:      [    0 , 0 , y^2 , 0 , -x , 0 ]
                    332:      [    0 , y , -x , 0 , 0 , 0 ]
                    333:      [    y*h , 0 , 0 , -x , 0 , 0 ]
                    334:      [    0 , 0 , 0 , 3*y*Dy , 0 , -2*Dx ]
                    335:    ]
                    336:    [
                    337:      [    1 , 0 , -y^2 , 0 , 0 ]
                    338:    ]
                    339:     [   ]
                    340:   ]
                    341:   [
                    342:     [    2*x*Dx , e_*x^2 , e_*x*y , 2*y*Dx*h , e_*y^3 , 3*y^2*Dy*h ]
                    343:     [    es*y^3 , es^2*y^2 , es*y , y*h , 3*es^3*y*Dy ]
                    344:     [    1 ]
                    345:   ]
                    346:   [
                    347:     [   ]
                    348:    [
                    349:     [
                    350:       [    1 , 4 ]
                    351:       [    y^3 , -x^2 ]
                    352:     ]
                    353:     [
                    354:       [    2 , 4 ]
                    355:       [    y^2 , -x ]
                    356:     ]
                    357:     [
                    358:       [    1 , 2 ]
                    359:       [    y , -x ]
                    360:     ]
                    361:     [
                    362:       [    0 , 3 ]
                    363:       [    y*h , -x ]
                    364:     ]
                    365:     [
                    366:       [    3 , 5 ]
                    367:       [    3*y*Dy , -2*Dx ]
                    368:     ]
                    369:    ]
                    370:    [
                    371:     [
                    372:       [    0 , 2 ]
                    373:       [    1 , -y^2 ]
                    374:     ]
                    375:    ]
                    376:     [   ]
                    377:   ]
                    378:   [   resolution $B$9$Y$-(B $BItJ,2C72(B e_ $B$O(B $B%Y%/%H%k@.J,$N%^!<%/(B.
                    379:     [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy ]
                    380:   ]
                    381:  ]
                    382:
                    383: $BN,(B
                    384: Processing [    1 , 3 ]    Strategy = 2
                    385:      1 $B$N(B 3 $BHVL\$N(B spair $B$N(B reduction $B$r=hM}Cf(B.
                    386:      In(7)=reductionTable:
                    387:     [[ 1 , 1 , 1 , 2 , 2 , 3 ]  , [    3 , 2 , 1 , 2 , 3 ]  , [    2 ]  ]
                    388:                                                    -- $B$3$l(B.
                    389: SpairAndReduction:
                    390: [    p and bases  , [    [    0 , 3 ]  , [    y*h , -x ]  ]  , [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , %[null] ]  ]
                    391: 0 $B$N(B 0 $BHVL\$H(B 3 $BHVL\(B $B$N(B spair $B$r7W;;$7$F(B, 0 $B%l%Y%k$N(B gb $B$G(B reduction.
                    392: [ 1 , 1 , 1 , 2 , 2 , 3 ] $B$K$"$k$h$&$K(B, strategy 3 $B0J30$O7W;;$:$_(B.
                    393: ( $B7W;;$7$F$J$$$b$N$O(B %[null] $B$H$J$C$F$k(B. )
                    394: [    level= , 1 ]
                    395: [    tower2= , [    [   ]  ]  ]   ( $B0lHV2<$J$N$G(B, tower $B$O$J$7$h(B. )
                    396: [    y*h , -es^3*x ]
                    397: [gi, gj] = [    2*x*Dx+3*y*Dy+6*h^2 , 2*y*Dx*h+3*x^2*Dy ]
                    398: 1
                    399: Reduce the element 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy
                    400: by  [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , %[null] ]
                    401: result is [    3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , 1 , [    0 , 0 , 0 , 0 , 0 , 0 ]  ]
                    402: vdegree of the original = -1
                    403: vdegree of the remainder = -1
                    404: [    3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , [    y*h , 0 , 0 , -x , 0 , 0 ]  , 3 , 5 , -1 , -1 ]
                    405:
                    406: In(11)=freeRes:
                    407: [    [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy ]  , [    %[null] , [    0 , 0 , y^2 , 0 , -x , 0 ]  , [    0 , -y , x , 0 , 1 , 0 ]  , [    -y*h , 0 , 0 , x , 0 , 1 ]  , %[null] ]  , [    %[null] ]  ]
                    408: $B$r$_$l$P$o$+$k$h$&$K(B, SlaScala $B$G(B, freeRes $B$K$3$N85$,(B [0,5] $B$K2C$((B
                    409: $B$i$l$?(B.
                    410:
                    411: $B<!$K(B SnextI $B$,(B SlaScala $B$h$j8F$P$l$F$3$N%(%i!<(B.
                    412:         i = SnextI(reductionTable_tmp,strategy,redundantTable,
                    413:                    skel,level,freeRes);
                    414: In(22)=reductionTable:
                    415: [    [    1 , 1 , 1 , 2 , 2 , 3 ]  , [    3 , 2 , 1 , 2 , 3 ]  , [    2 ]  ]
                    416: $B$J$N$G(B, $B:G8e(B $B$N(B 2 $B$,=hM}$5$l$k$O$:$@$,(B,
                    417: In(25)=skel[2]:
                    418: [    [    [    0 , 2 ]  , [    1 , -y^2 ]  ]  ]
                    419: $B$N$h$&$K(B, 0 $BHVL\$H(B, 2 $BHVL\$N(B spair.
                    420: $B$7$+$7(B,
                    421: In(26)=bases:
                    422: [    %[null] , [    0 , 0 , y^2 , 0 , -x , 0 ]  , [    0 , -y , x , 0 , 1 , 0 ]  , [    -y*h , 0 , 0 , x , 0 , 1 ]  , %[null] ]
                    423: $B$N$h$&$K(B, 0 $BHVL\$O(B strategy 3 $B$J$N$G(B, $B$^$@$b$H$^$C$F$$$J$$(B.
                    424:
                    425: reductionTable_tmp=[    2 ]
                    426: See also reductionTable, strategy, level,i
                    427: ERROR(sm): error operator : SnextI: bases[i] or bases[j] is null for all combinations.
                    428: --- Engine error or interrupt : In function : Error of class PrimitiveObject
                    429:
                    430: Type in Cleards() to exit the debug mode and Where() to see the stack trace.
                    431: In(7)=reductionTable:
                    432: [    [    1 , 1 , 1 , 2 , 2 , 3 ]  , [    3 , 2 , 1 , 2 , 3 ]  , [    2 ]  ]
                    433: In(8)=strategy:
                    434: 2
                    435: In(9)=level:
                    436: 2
                    437: $B$3$N;~E@$^$G$G$b$H$^$C$?(B basis
                    438:  [
                    439:    [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy ]
                    440:    [    %[null] , [    0 , 0 , y^2 , 0 , -x , 0 ]  , [    0 , -y , x , 0 , 1 , 0 ]  , [    -y*h , 0 , 0 , x , 0 , 1 ]  , %[null] ]
                    441:    [    %[null] ]
                    442:  ]
                    443:
                    444: -------------------------------------
                    445:
                    446: Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
                    447: a=Sminimal([x^2+y^2,x*y]);
                    448: $B$3$l$G$b;w$?$h$&$J%(%i!<$r$@$;$k(B.
                    449: $B$3$NJ}$,(B debug $B$7$d$9$$(B:
                    450: Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
                    451: a=Sminimal([x*y,x^2+y^2]);
                    452: $B$G$O%(%i!<$,$G$J$$$N$,IT;W5D(B.
                    453: pruneZero $B$,F0$$$F$J$$$N$,JQ(B.
                    454:
                    455: rf= [
                    456:   [
                    457:    [
                    458:      [    y^3 , 0 , -x^2 ]
                    459:      [    0 , y^2 , -x ]
                    460:      [    y , -x , 0 ]
                    461:    ]
                    462:    [
                    463:      [    1 , 0 , -y^2 ]
                    464:    ]
                    465:     [   ]
                    466:   ]
                    467:   [
                    468:     [    x^2 , x*y , y^3 ]
                    469:     [    y^3 , es*y^2 , y ]
                    470:     [    1 ]
                    471:   ]
                    472:   [
                    473:     [   ]
                    474:    [
                    475:     [
                    476:       [    0 , 2 ]
                    477:       [    y^3 , -x^2 ]
                    478:     ]
                    479:     [
                    480:       [    1 , 2 ]
                    481:       [    y^2 , -x ]
                    482:     ]
                    483:     [
                    484:       [    0 , 1 ]
                    485:       [    y , -x ]
                    486:     ]
                    487:    ]
                    488:    [
                    489:     [
                    490:       [    0 , 2 ]
                    491:       [    1 , -y^2 ]
                    492:     ]
                    493:    ]
                    494:     [   ]
                    495:   ]
                    496:   [
                    497:     [    x^2+y^2 , x*y , y^3 ]
                    498:   ]
                    499:  ]
                    500: [    0 , 0 ]
                    501: Processing [    0 , 0 ]    Strategy = 1
                    502: [    0 , 1 ]
                    503: Processing [    0 , 1 ]    Strategy = 1
                    504: [    1 , 2 ]
                    505: Processing [    1 , 2 ]    Strategy = 1
                    506: SpairAndReduction:
                    507: [    p and bases  , [    [    0 , 1 ]  , [    y , -x ]  ]  , [    x^2+y^2 , x*y , %[null] ]  ]
                    508: [    level= , 1 ]
                    509: [    tower2= , [    [   ]  ]  ]
                    510: [    y , -es*x ]
                    511: [gi, gj] = [    x^2+y^2 , x*y ]
                    512: 1
                    513: Reduce the element y^3
                    514: by  [    x^2+y^2 , x*y , %[null] ]
                    515: result is [    y^3 , 1 , [    0 , 0 , 0 ]  ]
                    516: vdegree of the original = -3
                    517: vdegree of the remainder = -3
                    518: [    y^3 , [    y , -x , 0 ]  , 2 , 2 , -3 , -3 ]
                    519: [    0 , 2 ]
                    520: Processing [    0 , 2 ]    Strategy = 2
                    521: [    1 , 1 ]
                    522: Processing [    1 , 1 ]    Strategy = 2
                    523: SpairAndReduction:
                    524: [    p and bases  , [    [    1 , 2 ]  , [    y^2 , -x ]  ]  , [    x^2+y^2 , x*y , y^3 ]  ]
                    525: [    level= , 1 ]
                    526: [    tower2= , [    [   ]  ]  ]
                    527: [    es*y^2 , -es^2*x ]
                    528: [gi, gj] = [    x*y , y^3 ]
                    529: 1
                    530: Reduce the element 0
                    531: by  [    x^2+y^2 , x*y , y^3 ]
                    532: result is [    0 , 1 , [    0 , 0 , 0 ]  ]
                    533: vdegree of the original = -4
                    534: vdegree of the remainder = %[null]
                    535: [    0 , [    0 , y^2 , -x ]  , 1 , -1 , -4 , %[null] ]
                    536: reductionTable_tmp=[    2 ]
                    537: See also reductionTable, strategy, level,i
                    538: ERROR(sm): error operator : SnextI: bases[i] or bases[j] is null for all combinations.
                    539: --- Engine error or interrupt : In function : Error of class PrimitiveObject
                    540:
                    541: Type in Cleards() to exit the debug mode and Where() to see the stack trace.
                    542: In(10)=reductionTable :
                    543: [    [    1 , 1 , 2 ]  , [    3 , 2 , 1 ]  , [    2 ]  ]
                    544: In(11)=bases:
                    545: [    %[null] , [    0 , y^2 , -x ]  , [    -y , x , 1 ]  ]
                    546: In(12)=  $B$3$l$O(B, [3, 2, 1]  $B$N85$N$&$A(B, 2,1 $B$,$b$H$^$C$F$$$k(B.
1.6     ! takayama  547: $B:G8e$N(B [ 2 ] $B$N7W;;$K(B 0 $BHVL\$,I,MW$G$3$l$,$^$@$J$$(B.
        !           548: $BMW$9$k$K(B 1 $BHVL\$H(B 3 $BHVL\$r>C$9(B operator [1, 0, -y^2]
        !           549:      [    y^3 , 0 , -x^2 ]
        !           550:      [    0 , y^2 , -x ]
        !           551:      [    y , -x , 0 ]
        !           552: $B$N(B reduction $B$,I,MW(B.
1.4       takayama  553:
1.5       takayama  554: -----------------------------------------
                    555: June 11, 2000 (Tue),  20:05
                    556: V-strict $B$+$I$&$+$r%A%'%C%/$9$k4X?t$r=q$-$?$$(B.
                    557: $B0BA4$K(B ring (schreyer order) $B$rDj5A$9$k4X?t$,M_$7$$(B.
                    558: $B0BA4$K(B parse $B$9$k4X?t$bM_$7$$(B.
                    559: $B%Y%/%H%k$H(B es $BI=8=$NJQ494X?t$b$$$k(B.
                    560:
                    561: AvoidTheSameRing == 1 $B$J$i(B, schreyer $B$N(B gbList $B$bJQ99$G$-$J$$$h$&$K(B
                    562: $B$9$Y$-$+!)(B
                    563: $B4XO"JQ?t(B:
                    564: needWarningForAvoidTheSameRing
                    565: isTheSameRing() :  ring $B$,F1$8$+(B check. pointer $B$G$J$/Cf?H$^$G$_$k(B.
                    566: see poly4.c.  $B$3$3$N%3%a%s%H$O;29M$K$J$k(B.
                    567: 3.If Schreyer = 1, then the system always generates a new ring.
                    568:
                    569: define_ring $B$K(B gbList $B$bEO$;$k$N(B?
                    570: ==> set_up_ring@ $B$r8+$k(B. grep set_up_ring ==>
                    571: primitive.c  KsetUpRing() grep KsetUpRing ==>
                    572: keyword gbListTower $B$,;H$($k$,(B, list $B$GM?$($J$$$H$$$1$J$$(B.
                    573: list $B$KJQ49$9$k$N$O(B, (list) dc.
                    574:
                    575: tparse $B$NI,MW$J$o$1(B?
                    576: ?? $B$*$b$$$@$;$J$$(B.
                    577:
                    578: ring_def $B$G(B ring (schreyer order) $B$rDj5A$9$k$H(B, $B7W;;$N$H$-$N(B
                    579: order $B$b(B tower $B$G$d$C$F$/$l$k$N(B?
                    580: $BB?J,(B NO.
                    581: grep ppAdd *.c ==>
                    582: poly2.c
                    583:   checkRing(f,g);
                    584:
                    585:   while (f != POLYNULL && g != POLYNULL) {
                    586:     /*printf("%s + %s\n",POLYToString(f,'*',1),POLYToString(g,'*',1));*/
                    587:     checkRing2(f,g); /* for debug */
                    588:     gt = (*mmLarger)(f,g);
                    589:
                    590:    mmLarger $B$OJQ$($F$J$$$h$&$K8+$($k(B.  checkRing $B$O%^%/%m(B.
                    591:
                    592: mmLarger_tower $B$O(B
                    593:   if (!(f->m->ringp->schreyer) || !(g->m->ringp->schreyer))
                    594:     return(mmLarger_matrix(f,g));
                    595: $B$H$J$C$F$k$N$G(B mmLarger_tower $B$r(B default $B$K$7$F$*$1$P?4G[$J$$$h$&$K8+$($k(B.
                    596:
                    597: ring_def $B$O@5$7$/F0$/(B?
                    598:
1.6     ! takayama  599: TODO:
        !           600: $B4X?t$N;EMM(B:   ( new.sm1 $B$^$?$O(B complex.sm1 $B$K$*$$$H$/(B )
1.5       takayama  601:   mmLarger $B$O(B tower $B$KJQ$($F$7$^$&(B.
                    602:   $BJQ?tL>(B, weight vector, $B%7%U%H%Y%/%H%k(B m $B$rM?$($k$H(B ring (with schreyer order)
                    603:   $B$r:n$k(B.   ==> weyl<m>,  weyl
1.6     ! takayama  604:   parser $B$O$H$/$K:n$kI,MW$,$J$$$h$&$K8+$($k$,(B...(tparse) ==> name
1.5       takayama  605:   $B%Y%/%H%k(B <---> es $BI=8=(B  cf. toVectors, [(toe_)  f] gbext ==> name
                    606:   $BE,@Z$J(B homogenization $B4X?t(B ==> homogenize<m>
                    607:   ord_w $B$N(B schreyer $BHG(B       ==> ord_w<m>
                    608:   init  $B$N(B schreyer $BHG(B       ==> init<m>
                    609:   gb_h, syz_h $B$NBP1~HG(B       ==> [ ii vv ww m] syz_h
                    610:   resolution $B$+$i(B shift vector $B$r7W;;$9$k4X?t(B.
                    611:
1.6     ! takayama  612:   $B7k2L$N(B check $B$r$9$k(B assert $B4X?t$bI,MW(B.
        !           613:
1.5       takayama  614: $B>e$N(B $B%7%U%H%Y%/%H%kBP1~HG$N4X?t$OEvJ,(B new.sm1 $B$X(B. $B$=$N$"$H(B complex.sm1 $B$X(B.
                    615:
                    616: cohom.sm1 $B$N(B interface $B4X?t$O(B cohom.k $B$X(B.
                    617: Help key word $B$O(B (Cohom.deRham) $B$_$?$$$K(B, . $B$G$/$.$C$F=q$/(B.
1.6     ! takayama  618:
        !           619: ----------------------
        !           620: $B%(%i!<$N860x$,$h$&$d$/$o$+$k(B:  June 14, 19:00
        !           621: Schreyer frame $B$NCJ3,$G(B syz $B$K(B 1 $B$,$"$k$H(B strategy $B$,(B
        !           622: $B$O$?$i$+$J$$(B.
        !           623:
        !           624: test13()  GKZ $B$N(B minimal free resolution.  2 $BEY<B9T$9$k$HJQ(B.
        !           625: grade $B$,JQ99$5$l$k$H(B, $BJQ$J$3$H$,$*$-$k$N$G(B,
        !           626: ScheckIfSchreyer() $B4X?t$G(B, $B$3$l$r(B scheck $B$9$k$3$H$K$7$?(B.
        !           627:   sm1(" (report) (mmLarger) switch_function /ss set ");
        !           628: $B$O$^$@$d$a$H$/(B. matrix $B$K$J$C$F$k$N$G(B.
        !           629:
        !           630: ------------------------------------------
        !           631: June 15, 2000
        !           632: TODO:
        !           633: 1.if (IdenfityIntegerAndUniversalNumber)  $B$N$H$-(B --- default
        !           634:   lt, gt, eq $B$G(B integer $B$H(B universalNumber $B$NHf3S$,$G$-$k$h$&$K$9$k(B.
        !           635:   rational $B$H$NHf3S$b2DG=$K$9$k(B.
        !           636:
        !           637: 2. sm1_push_int0 $B$KBP1~$9$k$3$H$r(B, sm1 $B$NB&$G$d$k(B.
        !           638:      $B%^%/%mL>(B  obj to_int --> Done.
        !           639:      weight_vector $B$N(B universalNumber ==> $B$^$@(B. $B%(%i!<$r$@$5$J$$$N$,$3$o$$(B.
        !           640:      s_weight_vector
        !           641:      weightv
        !           642:      ord_w
        !           643:      toVectors
        !           644:      define_ring
        !           645:      init
        !           646:      gkz
        !           647:
        !           648: -------------
        !           649: Schreyer skelton $B$,$I$&$7$F(B 1 $B$rMWAG$K$b$D$+$7$i$Y$k(B.

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