Annotation of OpenXM/src/k097/lib/minimal/minimal-note-ja.txt, Revision 1.8
1.8 ! takayama 1: $OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.7 2000/06/26 11:14:00 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$$(B.
640: s_weight_vector
641: weightv
642: ord_w
643: toVectors
644: define_ring
645: init
646: gkz
647:
648: -------------
649: Schreyer skelton $B$,$I$&$7$F(B 1 $B$rMWAG$K$b$D$+$7$i$Y$k(B.
1.7 takayama 650:
651: June 24 (Sat), 22:30 at Posthouse (Heathrow) www.posthouse-hotels.com
652: Sevilla $BBZ:_(B, Mega $B$b$h$&$d$/$*$o$j(B minimal resolution $B$N(B check $B$KLa$k(B.
653: resol1.c $B$K<!$N(B line $B$r2C$($?(B.
654: /* If isConstant(sv.a) is added, (x^3 - y^2 z^2) deRham stops
655: with an error. I've not yet understood the reason.
656: At Posthouse at Heathrow. June 24, 2000 */
657: if (isConstant(sv.b)) {
658: s->deleted = 1;
659: }
1.8 ! takayama 660: ===> $B$*$+$7$$$N$G:o=|(B.
1.7 takayama 661:
662: isConstant(sv.a) $B$,$J$$$H(B, $B$3$s$I$O(B,
663: Sminimal([x^2+y^2,x*y]); $B$,%(%i!<$G$H$^$k(B.
664: (x,y $B$N(B weight $B$O(B -1).
665: LaScala-Stillman $B$NO@J8$r$b$&0lEY$J$,$a$h$&(B.
666:
1.8 ! takayama 667: commit $B$9$Y$-(B: misc/mega2000 (cvs-misc add) Done.
! 668: OpenXM/src/kan96xx Done.
! 669: OpenXM/src/k097/lib/minimal Done.
! 670:
! 671: July 26.
! 672: resol.c $B$N(B schreyerSkelton0 $B$G(B, skelton $B$,(B minimal $B$K$J$k$h$&$K(B
! 673: $B%3!<%I$rA^F~(B.
! 674: $B%F%9%H$O(B
! 675: cd src/k097/lib/minimal
! 676: k0
! 677: load["minimal.k"];;
! 678: Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
! 679: Sminimal([x^2+y^2,x*y]);
! 680: $B$G(B.
! 681:
! 682: LaScala-Stillman $B$NO@J85U$G(B i<j $B$J$i(B e_i > e_j $B$H$J$k(B.
! 683: (order.c mmLarger_tower())
! 684:
! 685: $B%F%9%H(B 2.
! 686: cd src/k097/lib/minimal
! 687: k0
! 688: load["minimal-test.k"];;
! 689: v:
! 690: Sminimal(v);
! 691:
! 692: test11(); /* a = test_ann3("x^3-y^2*z^2"); */
! 693: test14(); /* gkz (1,2,3) */
! 694:
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