Annotation of OpenXM/src/k097/lib/minimal/example-ja.tex, Revision 1.4
1.4 ! takayama 1: % $OpenXM: OpenXM/src/k097/lib/minimal/example-ja.tex,v 1.3 2000/08/09 03:45:27 takayama Exp $
1.1 takayama 2: \documentclass[12pt]{jarticle}
3: \newtheorem{example}{Example}
4: \def\pd#1{ \partial_{#1} }
5: %% [2] should be replaced by \cite{....}
6:
7: \begin{document}
8: \section{$BNc(B}
9: $B2f!9$,(B $(u,v)$-$B6K>.<+M3J,2r$N9=@.$K6=L#$r$b$C$?F05!$N(B
10: $B0l$D$O(B, $D$ $B2C72(B $M$ $B$N@)8B%3%[%b%m%8$N7W;;$N8zN(2=$G$"$k(B.
11: [2] $B$G$O(B, $M$ $B$N(B Schreyer resolution $B$,(B $(-{\bf 1},{\bf 1})$
12: $B$KE,9g$7$?(B $M$ $B$N<+M3J,2r$G$"$k$3$H$r>ZL@$7(B,
13: $B$3$l$rMQ$$$?@)8B%3%[%b%m%8$N7W;;K!$rM?$($?(B.
14: $B$3$NJ}K!$rE,MQ$9$k$K$O(B
15: $(-w,w)$ $B$KE,9g$7$?(B $M$ $B$N<+M3J,2r$J$i$J$s$G$b$h$/(B,
16: Schreyer resolution $B$r$H$kI,A3@-$O$J$$(B.
17: [2] $B$NJ}K!$G$O(B, 1 $BE@$X$N@)8B$r7W;;$9$k$N$K<!85(B
18: $$O\left( (\mbox{ $B<+M3J,2r$N(B betti $B?t(B}) \times
19: \left(\mbox{$b$$B4X?t$N:GBg@0?t:,(B}\right)^n\right)$$
20: $B$N%Y%/%H%k6u4V$NJ#BN$N(B ${\rm Ker}/{\rm Im}$ $B$r(B
21: $B7W;;$9$kI,MW$,@8$8$k(B.
22: ( $BItJ,B?MMBN$X$N@)8B$K$O(B $D_m$, $(m < n)$ $B<+M32C72$NJ#BN(B
23: $B$N%3%[%b%m%8$r7W;;$9$kI,MW$,$"$k(B.)
24: $B$7$?$,$C$F(B, betti $B?t$,Bg$-$/$J$k$H(B, $B7W;;$9$Y$-(B $B%Y%/%H%k6u4V$N<!85$,(B
25: $BBg$-$/$J$j(B, $B%a%b%jITB-$r$^$M$$$F$$$?(B.
26: $(-w,w)$-$B6K>.<+M3J,2r$N(B betti $B?t$O(B Schreyer resolution $B$N(B betti $B?t$K(B
27: $BHf3S$7$F$+$J$j>.$5$/$$$^$^$G@)8B$,7W;;$G$-$J$+$C$?Nc$b7W;;$G$-$k$h$&$K(B
28: $B$J$C$?(B.
29:
30: Schreyer resolution $B$N(B betti $B?t$H(B $(-w,w)$-$B6K>.<+M3J,2r$N(B betti $B?t$r(B
31: $B$$$/$D$+$NNc$K$D$$$FHf3S$7$F$_$h$&(B.
32:
33: $BHf3S$NA0$K$$$/$D$+5-9f$HM=HwCN<1$rF3F~$9$k(B.
34: \begin{enumerate}
35: \item Schreyer resolution $B$N(B betti $B?t$O(B $(-w,w)$ $B$@$1$G$J$/(B
36: tie-breaking order $B$K$b0MB8$9$k(B.
37: $B0J2<(B tie-breaking order $B$H$7$F(B, graded reverse lexicographic order
38: $B$rMQ$$$k(B.
39: \item $BB?9`<0(B $f$ $B$N(B $b$-$B4X?t$N:G>.@0?t:,$r(B $-r$ $B$H$9$k$H$-(B
40: ${\rm Ann}(D f^{-1})$ $B$G(B
41: $1/f^r$ $B$rNm2=$9$k(B $D$ $B$N%$%G%"%k$N$"$k@8@.85$N=89g$r$"$i$o$9(B.
1.2 takayama 42: $B2<$N<BNc$N>l9g$G$O4X?t(B {\tt Sannfs(f,v)} $B$N=PNO$r$"$i$o$9(B.
1.1 takayama 43: \item $F(G)$ $B$G(B $G$ $B$N(B formal Laplace $BJQ49$r$"$i$o$9(B.
44: \item $F^h(G)$ $B$G(B $G$ $B$N(B formal Laplace $BJQ49$r(B homogenize $B$7$?$b$N$r$"$i$o$9(B.
45: \item Grothendieck $B$NHf3SDjM}$K$h$l$P(B
46: $I = F({\rm Ann} D f^{-1})$ $B$H$*$/$H$-(B $D/I$ $B$N86E@$X$N@)8B%3%[%b%m%8(B
47: $B$,6u4V(B $ {\bf C}^n \setminus V(f)$ $B$N(B ${\bf C}$-$B78?t%3%[%b%m%872$K0lCW$9$k(B.
48: ( "An algorithm for de Rham cohomology groups of the
49: complement of an affine variety via D-module computation",
50: Journal of pure and applied algebra, 139 (1999), 201--233. $B$r;2>H(B)
51: \end{enumerate}
52:
1.3 takayama 53: \begin{example} \rm \label{example:cusp}
1.1 takayama 54: %Prog: minimal-test.k test18()
55: $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2} \right) \right]$
56: $B$N>l9g(B.
57: $B%$%G%"%k(B $I$ $B$O(B
58: $$ -2x\pd{x}-3y\pd{y}+h^2 , -3y\pd{x}^2+2x\pd{y}h $$
59: $B$G@8@.$5$l$k(B.
60:
61: \begin{tabular}{|l|l|}
62: \hline
63: Resolution type & Betti numbers \\ \hline
1.2 takayama 64: Schreyer & 1, 4, 4, 1 \\ \hline
65: $(-{\bf 1},{\bf 1})$-minimal & 1, 2, 1 \\ \hline
66: minimal & 1, 2, 1 \\
1.1 takayama 67: \hline
68: \end{tabular}
69:
70: \noindent
71: $(-{\bf 1},{\bf 1})$-minimal resolution
72: {\footnotesize \begin{verbatim}
73: [
74: [
75: [ -2*x*Dx-3*y*Dy+h^2 ]
76: [ -3*y*Dx^2+2*x*Dy*h ]
77: ]
78: [
79: [ -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy ]
80: ]
81: ]
82: Degree shifts
83: [ [ 0 ] , [ 0 , 1 ] ]
84: \end{verbatim}}
85: Schreyer Resolution %%Prog: a=test18(); sm1_pmat(a[3]);
86: {\footnotesize \begin{verbatim}
87: [
88: [
89: [ -2*x*Dx-3*y*Dy+h^2 ]
90: [ -3*y*Dx^2+2*x*Dy*h ]
91: [ 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
92: [ 27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]
93: ]
94: [
95: [ 9*y^2*Dy+3*y*h^2 , 0 , 2*x , 1 ]
96: [ -4*x^2*Dy*h , 0 , -3*y*Dy+4*h^2 , Dx ]
97: [ 2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]
98: [ 3*y*Dx , -2*x , 1 , 0 ]
99: ]
100: [
101: [ -Dx , 1 , 2*x , 3*y*Dy-2*h^2 ]
102: ]
103: ]
104: \end{verbatim}}
105: \end{example}
106:
107: \begin{example} \rm
108: %Prog: minimal-test.k test17b()
109: $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2z^2} \right) \right]$
110: $B$N>l9g(B.
111:
112: \begin{tabular}{|l|l|}
113: \hline
114: Resolution type & Betti numbers \\ \hline
1.2 takayama 115: Schreyer & 1, 8, 16, 11, 2 \\ \hline
116: $(-{\bf 1},{\bf 1})$-minimal & 1, 4, 5, 2 \\ \hline
117: minimal & 1, 4, 5, 2 \\
1.1 takayama 118: \hline
119: \end{tabular}
120:
121: \noindent
122: $(-{\bf 1},{\bf 1})$-minimal resolution
123: {\footnotesize \begin{verbatim}
124: [
125: [
126: [ y*Dy-z*Dz ]
127: [ -2*x*Dx-3*z*Dz+h^2 ]
128: [ 2*x*Dy*Dz^2-3*y*Dx^2*h ]
129: [ 2*x*Dy^2*Dz-3*z*Dx^2*h ]
130: ]
131: [
132: [ 0 , 2*x*Dy^2*Dz-3*z*Dx^2*h , 0 , 2*x*Dx+3*z*Dz ]
133: [ 2*x*Dx+3*z*Dz-h^2 , y*Dy-z*Dz , 0 , 0 ]
134: [ 3*Dx^2*h , 0 , Dy , -Dz ]
135: [ 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 ]
136: [ 2*x*Dy*Dz , 0 , z , -y ]
137: ]
138: [
139: [ y , -2*x*Dy*Dz , 3*y*z , z , 2*x*Dx ]
140: [ Dz , -3*Dx^2*h , 2*x*Dx+3*y*Dy+3*z*Dz+6*h^2 , Dy , -3*Dy*Dz ]
141: ]
142: ]
143: Degree shifts
144: [ [ 0 ] , [ 0 , 0 , 2 , 2 ] , [ 2 , 0 , 3 , 2 , 1 ] ]
145: \end{verbatim}}
146: \end{example}
147:
148: \begin{example} \rm
149: %Prog: minimal-test.k test22();
150: $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3+y^3+z^3} \right) \right]$
151: $B$N>l9g(B.
152:
153: %% Uli Walther $B$N(B $BO@J8(B (MEGA 2000) $B$NNc$H(B betti $B?t$rHf3S$;$h(B.
154: \begin{tabular}{|l|l|}
155: \hline
156: Resolution type & Betti numbers \\ \hline
1.2 takayama 157: Schreyer & 1, 12, 44, 75, 70, 39, 13, 2 \\ \hline
158: $(-1,-2,-3,1,2,3)$-minimal & 1, 4, 5, 2 \\ \hline
159: minimal & 1, 4, 5, 2 \\
1.1 takayama 160: \hline
161: \end{tabular}
162:
163: \noindent
1.2 takayama 164: $(-1,-2,-3,1,2,3)$-minimal resolution
1.1 takayama 165: {\footnotesize \begin{verbatim}
1.2 takayama 166: [
167: [
168: [ x*Dx+y*Dy+z*Dz-3*h^2 ]
169: [ y*Dz^2-z*Dy^2 ]
170: [ x*Dz^2-z*Dx^2 ]
171: [ x*Dy^2-y*Dx^2 ]
172: ]
173: [
174: [ 0 , -x , y , -z ]
175: [ -x*Dz^2+z*Dx^2 , x*Dy , x*Dx+z*Dz-3*h^2 , z*Dy ]
176: [ -x*Dy^2+y*Dx^2 , -x*Dz , y*Dz , x*Dx+y*Dy-3*h^2 ]
177: [ -y*Dz^2+z*Dy^2 , x*Dx+y*Dy+z*Dz-2*h^2 , 0 , 0 ]
178: [ 0 , Dx^2 , -Dy^2 , Dz^2 ]
179: ]
180: [
181: [ -x*Dx+3*h^2 , y , -z , -x , 0 ]
182: [ -Dz^3-Dy^3 , -Dy^2 , Dz^2 , Dx^2 , -x*Dx-y*Dy-z*Dz ]
183: ]
184: ]
185: Degree shifts
186: [ [ 0 ] , [ 0 , 4 , 5 , 3 ] , [ 3 , 5 , 6 , 4 , 9 ] ]
1.1 takayama 187: \end{verbatim}}
188: \end{example}
189:
190:
191: \begin{example} \rm
192: %Prog: minimal-test.k test21();
193: $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2z^2+y^2+z^2} \right) \right]$
194: $B$N>l9g(B.
195:
196: \begin{tabular}{|l|l|}
197: \hline
198: Resolution type & Betti numbers \\ \hline
1.2 takayama 199: Schreyer & 1, 13, 43, 50, 21, 2 \\ \hline
200: $(-{\bf 1},{\bf 1})$-minimal & 1, 7, 10, 4 \\ \hline
201: minimal & 1, 7, 10, 4 \\
1.1 takayama 202: \hline
203: \end{tabular}
204:
205: \noindent
1.2 takayama 206: $f=x^3-y^2z^2+y^2+z^2$ $B$H$*$$$?>l9g(B,
207: $B6u4V(B ${\bf C}^3 \setminus V(f)$ $B$N(B
208: $B%3%[%b%m%872$N<!85$O(B
1.4 ! takayama 209: ${\rm dim}\, H^0 = 8$, ${\rm dim}\, H^1 = 0$,
! 210: ${\rm dim}\, H^2 = 1$, ${\rm dim}\, H^3 = 1$
1.2 takayama 211: $B$H$J$k(B.
212: $B$3$N>l9g(B $D/I$ $B$N(B
213: $b$-$B4X?t$N:GBg@0?t:,$O(B $2$ $B$H$J$j(B,
214: $B%3%[%b%m%8$r7W;;$9$k$?$a$K(B
1.4 ! takayama 215: $B9M$($k@~7A6u4V$NJ#BN$N<!85$O(B, $20, 28, 27, 11$ $B$G$"$k(B. %%Prog: Srestall_s.sm1
1.2 takayama 216: $B0lJ}(B Schreyer resolution $B$+$i%9%?!<%H$7$F(B,
1.1 takayama 217: $B@~7A6u4V$NJ#BN$r9M$($k$H(B, $B$=$N<!85$O(B
1.2 takayama 218: 130, 1078, 1667, 749, 40 $B$H$J$k(B. %%Prog: test21b()
1.1 takayama 219: \end{example}
220:
1.4 ! takayama 221: $B<!$K(B $(u,v)$-$B6K>.<+M3J,2r$H6K>.<+M3J,2r$,0[$J$kNc$r<($=$&(B.
! 222: \begin{example} \rm
! 223: %%Prog: minimal-test.k test24()
! 224: $BF1<!2=%o%$%kBe?t$N:8%$%G%"%k(B
! 225: $$I = D^{(h)}\cdot \{ h \pd{x} - x \pd{x} - y \pd{y},
! 226: h \pd{y} - x \pd{x} - y \pd{y} \} $$
! 227: $B$r9M$($k(B.
! 228:
! 229: \begin{tabular}{|l|l|}
! 230: \hline
! 231: Resolution type & Betti numbers \\ \hline
! 232: Schreyer & 1, 3, 3, 1 \\ \hline
! 233: $(-{\bf 1},{\bf 1})$-minimal & 1, 3, 2 \\ \hline
! 234: minimal & 1, 2, 1 \\
! 235: \hline
! 236: \end{tabular}
! 237:
! 238: \noindent
! 239: $(-{\bf 1},{\bf 1})$-minimal resolution
! 240: {\footnotesize \begin{verbatim}
! 241: [
! 242: [
! 243: [ Dx*h-x*Dx-y*Dy ]
! 244: [ Dy*h-x*Dx-y*Dy ]
! 245: [ x*Dx^2-x*Dx*Dy+y*Dx*Dy-y*Dy^2 ]
! 246: ]
! 247: [
! 248: [ x*Dx-x*Dy+y*Dy+x*h , -y*Dy-x*h , -h+x ]
! 249: [ -Dy+h , Dx-h , 1 ]
! 250: ]
! 251: ]
! 252: \end{verbatim}
! 253: } \noindent
! 254: $B$G$"$j(B, 1 $BHVL\$N(B syzygy $B$K(B
! 255: \verb# [-Dy+h, Dx-h, 1 ] #
! 256: $B$H(B $1$ $B$,=P8=$7$F$$$k(B.
! 257: $B<+M3J,2r$N<gIt(B (initial) $B$O0J2<$N$H$&$j(B.
! 258: {\footnotesize
! 259: \begin{verbatim}
! 260: [
! 261: [
! 262: [ Dx*h ]
! 263: [ Dy*h ]
! 264: [ x*Dx^2-x*Dx*Dy+y*Dx*Dy-y*Dy^2 ]
! 265: ]
! 266: [
! 267: [ x*Dx-x*Dy+y*Dy , -y*Dy , -h ]
! 268: [ -Dy , Dx , 0 ]
! 269: ]
! 270: ]
! 271: \end{verbatim}
! 272: }
! 273:
! 274: \noindent
! 275: $B0lJ}(B
! 276: minimal resolution %%Prog: test24b() minimal-test.k
! 277: $B$O(B
! 278: {\footnotesize \begin{verbatim}
! 279: [
! 280: [
! 281: [ Dx*h-x*Dx-y*Dy ]
! 282: [ Dy*h-x*Dx-y*Dy ]
! 283: ]
! 284: [
! 285: [ -Dy*h+x*Dx+y*Dy+h^2 , Dx*h-x*Dx-y*Dy-h^2 ]
! 286: ]
! 287: ]
! 288: \end{verbatim}
! 289: } \noindent
! 290:
! 291: \end{example}
! 292:
1.1 takayama 293: \begin{example} \rm
294: %Prog: minimal-test.k test20()
1.2 takayama 295: $I = D\cdot\{ x_1\pd{1}+2x_2\pd{2}+3x_3\pd{3} ,
296: \pd{1}^2-\pd{2}h,
297: -\pd{1}\pd{2}+\pd{3}h,
1.1 takayama 298: \pd{2}^2-\pd{1}\pd{3} \}
299: $ $B$N>l9g(B.
300: $B$3$l$O(B $A=(1,2,3)$, $\beta=0$ $B$KIU?o$9$k(B GKZ $BD64v2?7O$N(B
301: homogenization.
302:
303: \begin{tabular}{|l|l|}
304: \hline
305: Resolution type & Betti numbers \\ \hline
1.2 takayama 306: Schreyer & 1, 10, 25, 23, 8, 1 \\ \hline
307: $(-{\bf 1},{\bf 1})$-minimal & 1, 4, 5, 2 \\ \hline
308: minimal & 1, 4, 5, 2 \\
1.1 takayama 309: \hline
310: \end{tabular}
311:
312: \noindent
313: $(-{\bf 1},{\bf 1})$-minimal resolution
314: {\footnotesize \begin{verbatim}
315: [
316: [
317: [ x1*Dx1+2*x2*Dx2+3*x3*Dx3 ]
318: [ Dx1^2-Dx2*h ]
319: [ -Dx1*Dx2+Dx3*h ]
320: [ Dx2^2-Dx1*Dx3 ]
321: ]
322: [
323: [ Dx1*Dx2-Dx3*h , -x1*Dx2 , 2*x2*Dx2+3*x3*Dx3+3*h^2 , -x1*h ]
324: [ Dx1^2-Dx2*h , -x1*Dx1-3*x3*Dx3-2*h^2 , 2*x2*Dx1 , 2*x2*h ]
325: [ Dx2^2-Dx1*Dx3 , x1*Dx3 , x1*Dx2 , -2*x2*Dx2-3*x3*Dx3-4*h^2 ]
326: [ 0 , Dx3 , Dx2 , Dx1 ]
327: [ 0 , -Dx2 , -Dx1 , -h ]
328: ]
329: [
330: [ Dx2 , -Dx3 , -Dx1 , -2*x2*Dx2-3*x3*Dx3-4*h^2 , -x1*Dx2-2*x2*Dx3 ]
331: [ -Dx1 , Dx2 , h , -x1*h , -3*x3*Dx3-h^2 ]
332: ]
333: ]
334: Degree shifts
335: [ [ 0 ] , [ 0 , 2 , 2 , 2 ] , [ 2 , 2 , 2 , 3 , 3 ] ]
336: \end{verbatim}}
1.3 takayama 337: %%Prog:test23() of minimal-test.k
338: $B$3$N6K>.<+M3J,2r$O<B$O(B $B9TNs(B $(1,2,3)$ $B$G$-$^$k(B affine toric ideal
339: $B$N6K>.<+M3J,2r$N(B Koszul complex $B$K$J$C$F$k(B.
340: Gel'fand, Kapranov, Zelevinsky $B$K$h$C$FF3F~$5$l$?(B $D/I$
341: $B$N(B resolution $B$r<+A3$K1dD9$7$?<!$N(B 2 $B=EJ#BN$r9M$($h$&(B.
342: $$
343: \begin{array}{ccccccccc}
344: 0 & \longrightarrow & D^2 & \stackrel{d^1}{\longrightarrow}
345: & D^3 & \stackrel{d^2}{\longrightarrow}
346: & D & \longrightarrow & 0 \\
347: & & u^1 \downarrow &
348: & u^2 \downarrow &
349: & u^3 \downarrow & \\
350: 0 & \longrightarrow & D^2 & \stackrel{d^1}{\longrightarrow}
351: & D^3 & \stackrel{d^2}{\longrightarrow}
352: & D & \longrightarrow & 0
353: \end{array}
354: $$
355: $B$3$3$G$O(B, ($B8m2r$b$J$$$H;W$&$N$G(B) $D$ $B$GF1<!2=%o%$%kBe?t(B,
356: $d^i$ $B$G(B affine toric ideal $B$NF1<!2=$NB?9`<04D$G$N6K>.<+M3J,2r(B
357: $$ d^2 = \pmatrix{ \pd{1}^2 - \pd{2}^2 h \cr
358: -\pd{1} \pd{2} + \pd{3} h \cr
359: \pd{2}^2 - \pd{1} \pd{3} \cr }, \
360: d^1 = \pmatrix{ -\pd{2} & -\pd{1} & -h \cr
361: \pd{3} & \pd{2} & \pd{1} \cr }
362: $$
363: $B$r$"$i$o$9$b$N$H$9$k(B.
364: $B$^$?(B $\ell = x_1 \pd{1} + 2 x_2 \pd{2} + 3 x_3 \pd{3}$ $B$H$*$/$H$-(B $u^i$ $B$r(B
365: $B<!$N$h$&$K$-$a$k(B.
366: $$ u^1=\pmatrix{ \ell + 4 h^2 & 0 \cr
367: 0 & \ell+5 h^2 \cr}, \quad
368: u^2=\pmatrix{\ell+2 h^2 & 0 & 0 \cr
369: 0 & \ell + 3 h^2 & 0 \cr
370: 0 & 0 & \ell+ 4 h^2 \cr}, \quad
371: u^3 = \pmatrix{ \ell \cr}.
372: $$
373:
374: $B$3$N$H$-IU?o$9$k(B 1 $B=EJ#BN$O(B
375: $$L^1 \ni f \mapsto (-d^1(f), u^1(f)) \in L^2 \oplus L^1, $$
376: $$ L^2\oplus L^1 \ni (f,g)\mapsto (-d^2(f), u^2(f)+d^1(g)) \in L^3\oplus L^2,
377: $$
378: $$
379: L^3\oplus L^2 \ni (f,g)\mapsto u^3(f)+d^2(g) \in L^3.
380: $$
381: $B$G$"$?$($i$l$k(B.
382: $B$3$3$G(B $L^1 = D^2$, $L^2 = D^3$, $L^3 = D$ $B$G$"$k(B.
383: $B$3$N(B 1 $B=EJ#BN$N6qBN7A$O0J2<$N$H$&$j(B.
384: \footnotesize{
385: \begin{verbatim}
386: [
387: [
388: [ x1*Dx1+2*x2*Dx2+3*x3*Dx3 ]
389: [ Dx1^2-Dx2*h ]
390: [ -Dx1*Dx2+Dx3*h ]
391: [ Dx2^2-Dx1*Dx3 ]
392: ]
393: [
394: [ -Dx1^2+Dx2*h , x1*Dx1+2*x2*Dx2+3*x3*Dx3+2*h^2 , 0 , 0 ]
395: [ Dx1*Dx2-Dx3*h , 0 , x1*Dx1+2*x2*Dx2+3*x3*Dx3+3*h^2 , 0 ]
396: [ -Dx2^2+Dx1*Dx3 , 0 , 0 , x1*Dx1+2*x2*Dx2+3*x3*Dx3+4*h^2 ]
397: [ 0 , -Dx2 , -Dx1 , -h ]
398: [ 0 , Dx3 , Dx2 , Dx1 ]
399: ]
400: [
401: [ Dx2 , Dx1 , h , x1*Dx1+2*x2*Dx2+3*x3*Dx3+4*h^2 , 0 ]
402: [ -Dx3 , -Dx2 , -Dx1 , 0 , x1*Dx1+2*x2*Dx2+3*x3*Dx3+5*h^2 ]
403: ]
404: ]
405: \end{verbatim}
406: }
1.1 takayama 407: \end{example}
408:
409:
410:
411:
1.3 takayama 412: \section{$B<BAu(B}
413: $B$3$3$G$O(B
414: \begin{verbatim}
415: /* OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v 1.25
416: 2000/08/02 05:14:31 takayama Exp */
417: \end{verbatim}
418: $BHG$N(B {\tt minimal.k} $B$K=`5r$7$F<BAu$N35N,$r2r@b$9$k(B.
419:
420: $B<BAu$N@bL@$N$?$a$NNc$H$7$F%$%G%"%k(B
421: $$ I = D \cdot \{ -2x\pd{x}-3y\pd{y}+h^2, -3y\pd{x}^2+2x\pd{y}h \} $$
422: $B$N(B $(u,v) = (-1,-1,1,1)$-$B6K>.J,2r$N9=@.(B
423: $B$r9M$($h$&(B.
424: %%Prog: minimal-note-ja.txt 6/9 (Fri) $B$*$h$S0J8e$N(B bug fix $B$N5-O?$r;2>H(B.
425: %%$BNc$H$7$F(B, $B%$%G%"%k(B
426: %%$$ I = D \cdot \{ x^2 + y^2, x y \} $$
427: %%$B$N(B $(u,v) = (-1,-1,1,1)$-$B6K>.J,2r$N9=@.(B
428: %%$B$r9M$($h$&(B.
429: %%($B$3$N>l9g$OB?9`<04D$NF1<!<0$G@8@.$5$l$k$N$G(B, $BB?9`<04D$G$N(B
430: %% $B6K>.<+M3J,2r$N7W;;$HF1$8$3$H$K$J$k(B.)
431: $B$3$N>l9g(B,
432: $I$ $B$N%0%l%V%J4pDl(B $G$ $B$O(B
433: {\footnotesize
434: \begin{verbatim}
435: [
436: [ -2*x*Dx-3*y*Dy+h^2 ]
437: [ -3*y*Dx^2+2*x*Dy*h ]
438: [ 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
439: [ 27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]
440: ]
441: \end{verbatim}
442: } \noindent
443: $B$H$J$C$F$*$j(B,
444: Schreyer resolution $B$O(B
445: {\footnotesize
446: \begin{verbatim}
447: [
448: [
449: [ -2*x*Dx-3*y*Dy+h^2 ]
450: [ -3*y*Dx^2+2*x*Dy*h ]
451: [ 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
452: [ 27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]
453: ]
454: [
455: [ 9*y^2*Dy+3*y*h^2 , 0 , 2*x , 1 ]
456: [ -4*x^2*Dy*h , 0 , -3*y*Dy+4*h^2 , Dx ]
457: [ 2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]
458: [ 3*y*Dx , -2*x , 1 , 0 ]
459: ]
460: [
461: [ -Dx , 1 , 2*x , 3*y*Dy-2*h^2 ]
462: ]
463: ]
464: \end{verbatim}
465: } \noindent
466: $B$G$"$k(B. $1$ $B$,$?$/$5$s(B Schreyer resolution $B$NCf$K$O$"$k$3$H$K(B
467: $BCm0U(B. $1$ $B$O6K>.<+M3J,2r$K$OI,MW$J$$85$G$"$k$3$H$r0UL#$9$k(B.
468: $B6K>.<+M3J,2r$O(B, $BNc(B \ref{example:cusp} $B$K6qBN7A$r=q$$$F$*$$$?(B.
469:
470: \medbreak
471:
472:
473: $B$3$N<BAu$G$O6K>.<+M3J,2r$r(B LaScala $B$N%"%k%4%j%:%`$r$b$H$K$7$F(B
474: $B9=@.$9$k(B (LaScala and Stillman [??] $B$*$h$S?tM}2J3X$N5-;v(B ??? $B$r;2>H(B).
475:
476: $B$3$N%"%k%4%j%:%`$O4{CN$H$7$F(B, $B0c$$$N$_$r@bL@$7$h$&(B.
477: LaScala $B$N%"%k%4%j%:%`$O(B,
478: reduction $B$7$?$H$-$K(B $0$ $B$K$J$C$?>l9g(B, $B$=$N(B reduction $B$KIU?o$7$?(B
479: syzygy $B$r(B $B6K>.<+M3J,2r$N85$H$7(B,
480: reduction $B$7$?$H$-$K(B $0$ $B$K$J$i$J$+$C$?>l9g(B, $B$=$N85$r(B
481: $B%0%l%V%J4pDl$N85$H$7$F2C$((B, $BIU?o$7$?(B syzygy $B$O6K>.<+M3J,2r$G$OM>7W$J$b$N$H(B
482: $B$_$J$9(B.
483: $B$o$l$o$l$O(B $(u,v)$-$B6K>.$J<+M3J,2r$r$b$H$a$?$$(B.
484: $B$=$3$G>e$N<jB3$-$r<!$N$h$&$KJQ$($k(B.
485: \begin{center}
486: \begin{minipage}{10cm}
487: Reduction $B$7$?$H$-$K(B modulo $(u,v)$-$B%U%#%k%?!<$G(B $0$ $B$K$J$C$?>l9g(B,
488: $B$=$N(B reduction $B$KIU?o$7$?(B syzygy $B$r(B $B6K>.<+M3J,2r$N85$H$7(B, $B$5$i$K(B
489: $B$=$N85$,(B $0$ $B$G$J$1$l$P%0%l%V%J4pDl$K2C$($k(B.
490: reduction $B$7$?$H$-$K(B modulo $(u,v)$-$B%U%#%k%?!<$G(B $0$ $B$K$J$i$J$+$C$?>l9g(B,
491: $B$=$N85$r%0%l%V%J4pDl$N85$H$7$F2C$((B,
492: $BIU?o$7$?(B syzygy $B$O6K>.<+M3J,2r$G$OM>7W$J$b$N$H$_$J$9(B.
493: \end{minipage}
494: \end{center}
495:
496:
497: \bigbreak
498:
1.4 ! takayama 499: \noindent
1.3 takayama 500: {\tt minimal.k} $B$N%=!<%9%3!<%I$G$O$3$NItJ,$O<!$N$h$&$K$J$C$F$$$k(B.
501: {\footnotesize
502: \begin{verbatim}
503: def SlaScala(g,opt) {
504: ...
505: ...
506: f = SpairAndReduction(skel,level,i,freeRes,tower,ww);
507: if (f[0] != Poly("0")) {
508: place = f[3];
509: if (Sordinary) {
510: redundantTable[level-1,place] = redundant_seq;
511: redundant_seq++;
512: }else{
513: if (f[4] > f[5]) { ($B$$(B)
514: /* Zero in the gr-module */
515: Print("v-degree of [org,remainder] = ");
516: Println([f[4],f[5]]);
517: Print("[level,i] = "); Println([level,i]);
518: redundantTable[level-1,place] = 0;
519: }else{ ($B$m(B)
520: redundantTable[level-1,place] = redundant_seq;
521: redundant_seq++;
522: }
523: }
524: redundantTable_ordinary[level-1,place]
525: =redundant_seq_ordinary;
526: ...
527: ...
528: }
529: \end{verbatim}
530: }
531:
532: $B>/!9D9$/$J$k$,(B, $B$3$NItJ,$K$"$i$o$l$kJQ?t$N@bL@$r$7$h$&(B.
533:
534: LaScala $B$N%"%k%4%j%:%`$G$O(B, $B:G=i$K7W;;$9$Y$-(B S-pair $B$N7W;;<j=g(B,
535: $B$*$h$S(B Schreyer frame $B$r:n@.$9$k(B.
536: Schreyer frame $B$O(B Schreyer resolution $B$N(B initial $B$G$"$k(B.
537: $B$3$l$i$O$"$i$+$8$a(B
538: {\tt SresolutionFrameWithTower(g,opt);}
539: $B$G7W;;$5$l$F(B, {\tt tower} $B$*$h$S(B {\tt skel} $B$K3JG<$5$l$F$$$k(B.
540: $B$3$l$i$NJQ?t$NCM$O(B, $B4X?t(B {\tt Sminimal()} $B$NLa$jCM$H$7$F(B
541: $B8+$k$3$H$,$G$-$k(B.
542: $B4X?t(B {\tt Sminimal()} $B$NLa$jCM$,JQ?t(B $a$ $B$K3JG<$5$l$F$$$k$H$9$k$H(B,
543: {\tt a[0]} $B$,6K>.<+M3J,2r(B
544: {\tt a[3]} $B$,(B Schreyer $B<+M3J,2r(B($B$H$/$K(B {\tt a[3,0]} $B$,(B
545: $I$ $B$N%0%l%V%J4pDl(B),
546: {\tt a[4]} $B$,(B,
547: $B4X?t(B {\tt SlaScala()} $B$N(B
548: $BJQ?t(B {\tt [rf[0], tower, skel, rf[3]]} $B$NCM$G$"$k(B.
549: $B$7$?$,$C$F(B, {\tt tower} $B$O(B {\tt a[4,1]} $B$K3JG<$5$l$F$$$k(B.
550: $I$ $B$N>l9g$N(B {\tt tower} $B$O0J2<$N$H$&$j(B.
551: {\footnotesize
552: \begin{verbatim}
553: In(25)=sm1_pmat(a[4,1]);
554: [
555: [ -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ]
556: [ -9*y^2*Dy , -3*es^2*y*Dy , -3*es*y*Dy , -3*y*Dx ]
557: [ -Dx ]
558: ]
559: \end{verbatim}
560: } \noindent
561: $B$3$3$G(B ${\tt es}^i$ $B$O%Y%/%H%k$N(B $BBh(B $i$ $B@.J,$G$"$k$3$H$r$7$a$7$F$$$k(B.
562: $B$?$H$($P(B,
563: \verb# -3*es^2*y*Dy # $B$O(B
564: \verb# [0, 0, -3*y*Dy, 0] # $B$r0UL#$9$k(B.
565:
566: $BJQ?t(B
567: {\tt skel} $B$K$O(B
568: S-pair (sp) $B$N7W;;<j=g$,$O$$$C$F$$$k(B.
569: $I$ $B$N>l9g$K$O0J2<$N$H$&$j(B.
570: {\footnotesize
571: \begin{verbatim}
572: In(16)=sm1_pmat(a[4,2]);
573: [
574: [ ]
575: [
576: [
577: [ 0 , 2 ] G'[0] $B$H(B G'[2] $B$N(B sp $B$r7W;;(B (0)
578: [ -9*y^2*Dy , 2*x ]
579: ]
580: [
581: [ 2 , 3 ] G'[2] $B$H(B G'[3] $B$N(B sp $B$r7W;;(B (1)
582: [ -3*y*Dy , Dx ]
583: ]
584: [
585: [ 1 , 2 ] G'[1] $B$H(B G'[2] $B$N(B sp $B$r7W;;(B (2)
586: [ -3*y*Dy , Dx ]
587: ]
588: [
589: [ 0 , 1 ] G'[0] $B$H(B G'[1] $B$N(B sp $B$r7W;;(B (3)
590: [ -3*y*Dx , 2*x ]
591: ]
592: ]
593: [
594: [
595: [ 0 , 3 ] G''[0] $B$H(B G''[3] $B$N(B sp $B$r7W;;(B
596: [ -Dx , 3*y*Dy ]
597: ]
598: ]
599: [ ]
600: ]
601: \end{verbatim}
602: } \noindent
603: $B$3$3$G(B $G'$ $B$O(B Schreyer order $B$GF@$i$l$?(B $G$ $B$N(B syzygy $B$N@8@.85(B,
604: $G''$ $B$O(B Schreyer order $B$GF@$i$l$?(B $G'$ $B$N(B syzygy $B$N@8@.85$r$"$i$o$9(B.
605: $B$?$H$($P>e$NNc$G$O(B,
606: $G'[0]$ $B$O(B
607: $G[0]$ $B$H(B $G[2]$ $B$N(B sp $B$N7W;;$K$h$jF@$i$l$?(B syzygy,
608: $G'[1]$ $B$O(B
609: $G[2]$ $B$H(B $G[3]$ $B$N(B sp $B$N7W;;$K$h$jF@$i$l$?(B syzygy,
610: ...
611: $B$r0UL#$9$k(B.
612:
613: {\footnotesize
614: \begin{verbatim}
615: f = SpairAndReduction(skel,level,i,freeRes,tower,ww);
616: \end{verbatim}
617: } \noindent
618: $B$G$O(B {\tt skel[level,i]} $B$K3JG<$5$l$?(B
619: S-pair $B$r7W;;$7$F(B, {\tt freeRes[level-1]} $B$G(B reduction $B$r$*$3$J$&(B.
620: Reduction $B$N$?$a$N(B Schreyer order $B$O(B \\
621: {\tt StowerOf(tower,level-1)} $B$rMQ$$$k(B.
622: $B$?$H$($P(B, ${\tt [level,i] = [1,3]}$ $B$N$H$-$K(B
623: $B4X?t(B {\tt SpairAndReduction} $B$G(B
624: $B$I$N$h$&$J7W;;$,$J$5$l$F$$$k$+(B $I$ $B$N>l9g$K$_$F$_$h$&(B.
625:
626: {\tt SpairAndReduction} $B$N<B9T;~(B
627: $B$K<!$N$h$&$J%a%C%;!<%8$,$G$F$/$k(B.
628: {\footnotesize
629: \begin{verbatim}
630: reductionTable= [
631: [ 1 , 2 , 3 , 4 ]
632: [ 3 , 4 , 3 , 2 ]
633: [ 3 ]
634: ]
635: [ 0 , 0 ]
636: Processing [level,i]= [ 0 , 0 ] Strategy = 1
637: [ 0 , 1 ]
638: Processing [level,i]= [ 0 , 1 ] Strategy = 2
639: [ 1 , 3 ]
640: Processing [level,i]= [ 1 , 3 ] Strategy = 2
641: SpairAndReduction:
642: [ p and bases , [ [ 0 , 1 ] , [ -3*y*Dx , 2*x ] ] ,
643: [-2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ] ]
644: [ level= , 1 ]
645: [ tower2= , [ [ ] ] ]
646: [ -3*y*Dx , 2*es*x ]
647: [gi, gj] = [ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h ]
648: 1
649: Reduce the element 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h
650: by [ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ]
651: result is [ 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , 1 , [ 0 , 0 , 0 , 0 ] ]
652: vdegree of the original = 0
653: vdegree of the remainder = 0
654: [ 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ,
655: [ -3*y*Dx , 2*x , 0 , 0 ] , 3 , 2 , 0 , 0 ]
656: \end{verbatim}
657: } \noindent
658: $B:G=i$KI=<($5$l$k(B {\tt reductionTable} $B$N0UL#$O$"$H$G@bL@$9$k(B.
659: $B<!$N9T$KCmL\$7$h$&(B. $B$3$3$G$O(B {\tt skel[0,4]} $B$N(B S-pair
660: $B$r7W;;$7$F(Breduction $B$7$F$$$k(B.
661: {\footnotesize
662: \begin{verbatim}
663: SpairAndReduction:
664: [ p and bases , [ [ 0 , 1 ] , [ -3*y*Dx , 2*x ] ] ,
665: [-2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ] ]
666: \end{verbatim}
667: } \noindent
668: {\tt [0, 1]} $B$O(B $G'[0]$ $B$H(B $G'[1]$ $B$N(B sp $B$r7W;;(B
669: $B$;$h$H$$$&0UL#$G$"$k(B.
670: ${\tt level} = 0$ $B$G4{$K$b$H$^$C$F$$$k(B $B%V%l%V%J4pDl$O(B
671: $G[0]$ $B$H(B $G[1]$ $B$N$_$G$"$j(B,
672: $B$=$l$i$O$=$l$>$l(B,
673: \verb# -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h #
674: $B$G$"$k(B.
675: {\tt SpairAndReduction} $B$O(B $G[0]$, $G[1]$ $B$N$_$rMQ$$$F(B,
676: S-pair \\
677: \verb# 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h #
678: $B$r(B reduction $B$9$k(B.
679: $B7k6I(B reduction $B$N7k2L$O(B 0 $B$G$O$J$/$F(B, \\
680: \verb# 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h #
681: $B$H$J$k(B.
682: LaScala $B$N%"%k%4%j%:%`$N(B 2 $BG\$*F@%7%9%F%`$G(B,
683: $B$3$l$,?7$7$$%0%l%V%J4pDl$N85(B {\tt G[place]} $B$H$J$j(B,
684: reduction $B$N2aDx$h$j(B syzygy $B$bF@$i$l$k(B.
685:
686: $B$5$F(B, $(u,v)$-$B6K>.J,2r$r:n$k$K$O(B, reduction $B$7$?M>$j$,(B
687: $(u,v)$-$B%U%#%k%?!<$G(B modulo $B$7$F(B $0$ $B$+$I$&$+D4$Y$J$$$H$$$1$J$$(B.
688: $B$3$N$?$a(B,
689: $B4X?t(B {\tt Sdegree()} $B$rMQ$$$F(B, reduction $B$9$kA0$N85(B, $B$*$h$SM>$j$N(B
690: $B%7%U%HIU$-(B $(u,v)$-order $B$r7W;;$9$k(B.
691: $B$3$NNc$G$O(B, $BN>J}$H$b(B $0$ $B$G$"$k(B.
692: {\footnotesize
693: \begin{verbatim}
694: vdegree of the original = 0
695: vdegree of the remainder = 0
696: \end{verbatim}
697: }
698: $B$7$?$,$C$F(B, modulo $(u,v)$-$B%U%#%k%?!<$G$b(B $0$ $B$G$J$$(B.
1.1 takayama 699:
1.3 takayama 700: $B=`Hw@bL@$,$*$o$C$?(B. $B:G=i$N%W%m%0%i%`(B {\tt SlaScala()} $B$N@bL@$KLa$k(B.
701: {\tt SpairAndReduction()} $B$NLa$jCM(B
702: {\tt f[0]} $B$K$O(B, reduction $B$7$?M>$j(B,
703: {\tt f[4]}, {\tt f[5]} $B$K$O(B,
704: reduction $B$9$kA0$N85(B, $B$*$h$SM>$j$N(B
705: $B%7%U%HIU$-(B $(u,v)$-order $B$,3JG<$5$l$F$$$k(B.
706: $B$3$NNc$N>l9g$K$O(B ($B$m(B) $B$N>l9g$,<B9T$5$l$F(B,
707: $BIU?o$7$?(B syzygy $B$O(B $B6K>.<+M3J,2r$K$OITMW$J$b$N$H$7$F(B
708: {\tt redundantTable} $B$KEPO?$5$l$k(B:
709: {\footnotesize
710: \begin{verbatim}
711: redundantTable[level-1,place] = redundant_seq;
712: \end{verbatim}
713: } \noindent
714: $BM>$j(B {\tt f[0]} $B$O(B, laScala $B$N%"%k%4%j%:%`$N(B 2 $BG\$*F@8=>]$GF@$i$l$?(B,
715: $B?7$7$$%V%l%V%J4pDl$N85$G$"$k$,(B, $B$3$l$rJ]B8$9$Y$->l=j$N%$%s%G%C%/%9$O(B,
716: $BLa$jCM(B {\tt f[3]}({\tt place}) $B$K3JG<$5$l$F$$$k(B:
717: {\footnotesize
718: \begin{verbatim}
719: bases[place] = f[0];
720: freeRes[level-1] = bases;
721: reducer[level-1,place] = f[1];
722: \end{verbatim}
723: } \noindent
724: $B$3$N(B reduction $B$GF@$i$l$?(B syzygy ($B$NK\<AE*ItJ,(B)$B$O(B,
725: $BJQ?t(B {\tt reducer} $B$KEPO?$5$l$k(B.
726: $B0J>e$G(B $(u,v)$-$B6K>.<+M3J,2rFCM-$N=hM}$NItJ,$N2r@b$r=*$($k(B.
727:
728:
729: \bigbreak
730: $B0J2<$G$O(B, LaScala $B$N%"%k%4%j%:%`$N$o$l$o$l$N<BAu$N35N,$HLdBjE@$r(B
731: $B=R$Y$k(B.
732:
733: $B$^$:(B, $BJQ?t(B
734: {\tt reductionTable} $B$N0UL#$r@bL@$7$h$&(B.
735: LaScala $B$N%"%k%4%j%:%`$G$O(B,
736: {\tt level - Sdegree(s)}
737: $B$N>.$5$$(B S-pair $B$+$i7W;;$7$F$$$/(B.
738: $B4X?t(B {\tt Sdegree} $B$O<!$N$h$&$K:F5"E*$KDj5A$5$l$F$$$k(B.
739: {\footnotesize
740: \begin{verbatim}
741: /* f is assumed to be a monomial with toes. */
742: def Sdegree(f,tower,level) {
743: local i,ww, wd;
744: /* extern WeightOfSweyl; */
745: ww = WeightOfSweyl;
746: f = Init(f);
747: if (level <= 1) return(StotalDegree(f));
748: i = Degree(f,es);
749: return(StotalDegree(f)+Sdegree(tower[level-2,i],tower,level-1));
750: }
751: \end{verbatim}
752: } \noindent
753: $B$3$3$G(B {\tt StotalDegree(f)} $B$O(B $f$ $B$NA4<!?t$G$"$k(B.
754:
755: \noindent
756: $B$5$F(B, LaScala $B$N%"%k%4%j%:%`$G$O(B,
757: Resolution $B$r2<$+$i=gHV$K7W;;$7$F$$$/$N$G$O$J$$(B.
758: $B$3$l$,K\<AE*$JE@$G$"$k(B.
759: $B$3$N=gHV$OJQ?t(B {\tt reductionTable} $B$K$O$C$F$$$k(B.
760: $I$ $B$NNc$G$O(B
761: {\footnotesize
762: \begin{verbatim}
763: reductionTable= [
764: [ 1 , 2 , 3 , 4 ]
765: [ 3 , 4 , 3 , 2 ] skel[0] $B$KBP1~(B
766: [ 3 ] skel[1] $B$KBP1~(B
767: ]
768: \end{verbatim}
769: } \noindent
770: $B$H$J$k(B.
771:
772: $B8=:_$N<BAu$G$N7W;;B.EY(B, $B%a%b%j;HMQNL$N%\%H%k%M%C%/$r(B
773: $B;XE&$7$F$*$/(B.
774: LaScala $B$N%"%k%4%j%:%`$G$O(B, Schreyer Frame $B$r9=@.$7$F$+$i(B,
775: $B6K>.<+M3J,2r$r9=@.$9$k(B.
776: $B2<5-$N%W%m%0%i%`$NJQ?t(B {\tt redundantTable[level,q]} $B$K$O(B,
777: $BBP1~$9$k(B syzygy $B$H(B $B%0%l%V%J4pDl$N85$,2?2sL\$N(B reduction $B$G@8@.(B
778: $B$5$l$?$+$N?t$,$O$$$C$F$$$k(B.
779: $B6K>.<+M3J,2r$N9=@.$G$O(B, $B:G8e$N(B reduction $B$N(B syzygy $B$+$i;O$a$F(B,
780: Schreyer resolution $B$+$i6K>.<+M3J,2r$K$H$C$FM>J,$J85$r<h$j=|$$$F(B
781: $B$$$/(B
782: ({\tt seq} $B$r(B $1$ $B$E$D8:$i$7$F$$$/(B).
783: {\footnotesize
1.1 takayama 784: \begin{verbatim}
1.3 takayama 785: def Sminimal(g,opt) {
786:
787: ....
788:
789: while (seq > 1) {
790: seq--;
791: for (level = 0; level < maxLevel; level++) {
792: betti = Length(freeRes[level]);
793: for (q = 0; q<betti; q++) {
794: if (redundantTable[level,q] == seq) {
795: Print("[seq,level,q]="); Println([seq,level,q]);
796: if (level < maxLevel-1) {
797: bases = freeRes[level+1];
798: dr = reducer[level,q];
799: dr[q] = -1;
800: newbases = SnewArrayOfFormat(bases);
801: betti_levelplus = Length(bases);
802: /*
803: bases[i,j] ---> bases[i,j]+bases[i,q]*dr[j]
804: */
805: for (i=0; i<betti_levelplus; i++) {
806: newbases[i] = bases[i] + bases[i,q]*dr;
807: }
808: ....
809: }
810: ....
811: }
812: }
813: }
814: }
815: ....
816: }
1.1 takayama 817: \end{verbatim}
1.3 takayama 818: } \noindent
819: $BLdBj$O(B,
820: $B6K>.<+M3J,2r<+BN$O$A$$$5$/$F$b(B, Schreyer Frame $B$,5pBg(B ($10000$ $BDxEY$N(B
821: betti $B?t(B) $B$H$J$k$3$H$bB?$$>l9g$,$"$k$3$H$G$"$k(B.
822: $B2<$NJQ?t(B {\tt bases} $B$K(B, Schreyer resolution $B$N(B {\tt level} $B<!$N(B
823: syzygy $B$r$$$l$F$$$k(B. Schreyer Frame $B$K(B $10000$ $BDxEY$N(B betti
824: $B?t$,$"$i$o$l$k$H$3$NJQ?t$O(B $B%5%$%:(B $10000$ $BDxEY$NG[Ns$H$J$k(B.
825: $B$5$i$K(B, Schreyer $BJ,2r$+$i6K>.<+M3J,2r$N$?$a$KITMW$J85$r$H$j$N$>$$$?(B
826: $BJ,2r$r:n$k$?$a$K(B\\
827: \verb# newbases[i] = bases[i] + bases[i,q]*dr; # \\
828: $B$J$k>C5n$r$*$3$J$$(B, $0$ $B$GKd$a$i$l$?Ns$^$?$O(B $0$ $B$GKd$a$i$l$?9T$r@8@.$7$F$$$k(B.
829: $B$3$NItJ,$,(B, $B%a%b%j$N;HMQ$r05Gw$7$F$*$j(B, $B7W;;;~4V$b$D$+$C$F$$$k(B.
830:
1.1 takayama 831:
832:
833: \end{document}
FreeBSD-CVSweb <freebsd-cvsweb@FreeBSD.org>
![[BACK]](/icons/cvsweb/back.gif){kind=icon}
Return to example-ja.tex CVS log ![[TXT]](/icons/cvsweb/text.gif){kind=icon}
![[DIR]](/icons/cvsweb/dir.gif){kind=icon}
Up to [local] / OpenXM / src / k097 / lib / minimal