Annotation of OpenXM/src/k097/lib/minimal/minimal-note-ja.txt, Revision 1.4
1.4 ! takayama 1: $OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.3 2000/06/08 08:37:53 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.
! 547: [ 2 ] $B$N7W;;$K(B 0 $BHVL\$,I,MW$G$3$l$,$^$@$J$$(B.
! 548:
! 549:
! 550:
1.3 takayama 551:
FreeBSD-CVSweb <freebsd-cvsweb@FreeBSD.org>
![[BACK]](/icons/cvsweb/back.gif){kind=icon}
Return to minimal-note-ja.txt CVS log ![[TXT]](/icons/cvsweb/text.gif){kind=icon}
![[DIR]](/icons/cvsweb/dir.gif){kind=icon}
Up to [local] / OpenXM / src / k097 / lib / minimal