Annotation of OpenXM/src/k097/lib/minimal/example-ja.tex, Revision 1.2
1.2 ! takayama 1: % $OpenXM: OpenXM/src/k097/lib/minimal/example-ja.tex,v 1.1 2000/08/02 03:23:36 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:
53: \begin{example} \rm
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
! 209: ${\rm dim}\, H^i = 1$, $(i=0, 1)$,
! 210: ${\rm dim}\, H^i = 0$, $(i=2, 3)$,
! 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
! 215: $B9M$($k@~7A6u4V$NJ#BN$N<!85$O(B, $10, 12, 9, 4$ $B$G$"$k(B. %%Prog: Srestall.sm1
! 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:
221: \begin{example} \rm
222: %Prog: minimal-test.k test20()
1.2 ! takayama 223: $I = D\cdot\{ x_1\pd{1}+2x_2\pd{2}+3x_3\pd{3} ,
! 224: \pd{1}^2-\pd{2}h,
! 225: -\pd{1}\pd{2}+\pd{3}h,
1.1 takayama 226: \pd{2}^2-\pd{1}\pd{3} \}
227: $ $B$N>l9g(B.
228: $B$3$l$O(B $A=(1,2,3)$, $\beta=0$ $B$KIU?o$9$k(B GKZ $BD64v2?7O$N(B
229: homogenization.
230:
231: \begin{tabular}{|l|l|}
232: \hline
233: Resolution type & Betti numbers \\ \hline
1.2 ! takayama 234: Schreyer & 1, 10, 25, 23, 8, 1 \\ \hline
! 235: $(-{\bf 1},{\bf 1})$-minimal & 1, 4, 5, 2 \\ \hline
! 236: minimal & 1, 4, 5, 2 \\
1.1 takayama 237: \hline
238: \end{tabular}
239:
240: \noindent
241: $(-{\bf 1},{\bf 1})$-minimal resolution
242: {\footnotesize \begin{verbatim}
243: [
244: [
245: [ x1*Dx1+2*x2*Dx2+3*x3*Dx3 ]
246: [ Dx1^2-Dx2*h ]
247: [ -Dx1*Dx2+Dx3*h ]
248: [ Dx2^2-Dx1*Dx3 ]
249: ]
250: [
251: [ Dx1*Dx2-Dx3*h , -x1*Dx2 , 2*x2*Dx2+3*x3*Dx3+3*h^2 , -x1*h ]
252: [ Dx1^2-Dx2*h , -x1*Dx1-3*x3*Dx3-2*h^2 , 2*x2*Dx1 , 2*x2*h ]
253: [ Dx2^2-Dx1*Dx3 , x1*Dx3 , x1*Dx2 , -2*x2*Dx2-3*x3*Dx3-4*h^2 ]
254: [ 0 , Dx3 , Dx2 , Dx1 ]
255: [ 0 , -Dx2 , -Dx1 , -h ]
256: ]
257: [
258: [ Dx2 , -Dx3 , -Dx1 , -2*x2*Dx2-3*x3*Dx3-4*h^2 , -x1*Dx2-2*x2*Dx3 ]
259: [ -Dx1 , Dx2 , h , -x1*h , -3*x3*Dx3-h^2 ]
260: ]
261: ]
262: Degree shifts
263: [ [ 0 ] , [ 0 , 2 , 2 , 2 ] , [ 2 , 2 , 2 , 3 , 3 ] ]
264: \end{verbatim}}
265: %% $B$3$N(B resolution $B$O<B$O(B, toric $B$N(B resolution $B$N(B Koszul complex $B$K$J$C$F$k(B
266: %% $B$O$:(B.
267: \end{example}
268:
269:
270:
271:
272: $(-w,w)$-$B6K>.<+M3J,2r$H(B $B6K>.<+M3J,2r$,$3$H$J$kNc$r$5$,$7$F$$$k$,(B
273: $B$3$l$O$^$@8+$D$+$C$F$$$J$$(B.
274:
275: \section{$B<BAu(B}
276: $B$3$3$G$O(B
277: \begin{verbatim}
278: /* OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v
279: 1.23 2000/08/01 08:51:03 takayama Exp */
280: \end{verbatim}
281: $BHG$N(B {\tt minimal.k} $B$K=`5r$7$F<BAu$N35N,$r2r@b$9$k(B.
282:
283: $B$^$@=q$$$F$J$$(B.
284:
285: \end{document}
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