Annotation of OpenXM/src/k097/lib/minimal/example-ja.tex, Revision 1.1
1.1 ! takayama 1: % $OpenXM$
! 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.
! 42: $B2<$N<BNc$G$O4X?t(B {\tt Sannfs(f,v)} $B$N=PNO$r$"$i$o$9(B.
! 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
! 64: Schreyer & 2, 1 \\ \hline
! 65: $(-{\bf 1},{\bf 1})$-minimal & 4, 4, 1 \\ \hline
! 66: minimal & 2, 1 \\
! 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
! 115: Schreyer & 4, 5, 2 \\ \hline
! 116: $(-{\bf 1},{\bf 1})$-minimal & 8, 16, 11, 2 \\ \hline
! 117: minimal & 4, 5, 2 \\
! 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
! 157: Schreyer & \\ \hline
! 158: $(-{\bf 1},{\bf 1})$-minimal & \\ \hline
! 159: minimal & \\
! 160: \hline
! 161: \end{tabular}
! 162:
! 163: \noindent
! 164: $(-{\bf 1},{\bf 1})$-minimal resolution
! 165: {\footnotesize \begin{verbatim}
! 166:
! 167: \end{verbatim}}
! 168: \end{example}
! 169:
! 170:
! 171: \begin{example} \rm
! 172: %Prog: minimal-test.k test21();
! 173: $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2z^2+y^2+z^2} \right) \right]$
! 174: $B$N>l9g(B.
! 175:
! 176: \begin{tabular}{|l|l|}
! 177: \hline
! 178: Resolution type & Betti numbers \\ \hline
! 179: Schreyer & \\ \hline
! 180: $(-{\bf 1},{\bf 1})$-minimal & \\ \hline
! 181: minimal & \\
! 182: \hline
! 183: \end{tabular}
! 184:
! 185: \noindent
! 186: $(-{\bf 1},{\bf 1})$-minimal resolution
! 187: {\footnotesize \begin{verbatim}
! 188:
! 189: \end{verbatim}}
! 190: $B%3%[%b%m%872$O(B ... $B$H$J$k(B.
! 191: $B9M$($k@~7A6u4V$NJ#BN$N<!85$O(B, ...
! 192: Schreyer resolution $B$+$i%9%?!<%H$7$F(B,
! 193: $B@~7A6u4V$NJ#BN$r9M$($k$H(B, $B$=$N<!85$O(B
! 194: ... $B$H$J$k(B.
! 195: \end{example}
! 196:
! 197: \begin{example} \rm
! 198: %Prog: minimal-test.k test20()
! 199: $I = D\cdot\{ x_1*\pd{1}+2x_2\pd{2}+3x_3\pd{3} ,
! 200: \pd{1}^2-\pd{2}*h,
! 201: -\pd{1}\pd{2}+\pd{3}*h,
! 202: \pd{2}^2-\pd{1}\pd{3} \}
! 203: $ $B$N>l9g(B.
! 204: $B$3$l$O(B $A=(1,2,3)$, $\beta=0$ $B$KIU?o$9$k(B GKZ $BD64v2?7O$N(B
! 205: homogenization.
! 206:
! 207: \begin{tabular}{|l|l|}
! 208: \hline
! 209: Resolution type & Betti numbers \\ \hline
! 210: Schreyer & 4, 5, 2 \\ \hline
! 211: $(-{\bf 1},{\bf 1})$-minimal & 10, 25, 23, 8, 1 \\ \hline
! 212: minimal & 4, 5, 2 \\
! 213: \hline
! 214: \end{tabular}
! 215:
! 216: \noindent
! 217: $(-{\bf 1},{\bf 1})$-minimal resolution
! 218: {\footnotesize \begin{verbatim}
! 219: [
! 220: [
! 221: [ x1*Dx1+2*x2*Dx2+3*x3*Dx3 ]
! 222: [ Dx1^2-Dx2*h ]
! 223: [ -Dx1*Dx2+Dx3*h ]
! 224: [ Dx2^2-Dx1*Dx3 ]
! 225: ]
! 226: [
! 227: [ Dx1*Dx2-Dx3*h , -x1*Dx2 , 2*x2*Dx2+3*x3*Dx3+3*h^2 , -x1*h ]
! 228: [ Dx1^2-Dx2*h , -x1*Dx1-3*x3*Dx3-2*h^2 , 2*x2*Dx1 , 2*x2*h ]
! 229: [ Dx2^2-Dx1*Dx3 , x1*Dx3 , x1*Dx2 , -2*x2*Dx2-3*x3*Dx3-4*h^2 ]
! 230: [ 0 , Dx3 , Dx2 , Dx1 ]
! 231: [ 0 , -Dx2 , -Dx1 , -h ]
! 232: ]
! 233: [
! 234: [ Dx2 , -Dx3 , -Dx1 , -2*x2*Dx2-3*x3*Dx3-4*h^2 , -x1*Dx2-2*x2*Dx3 ]
! 235: [ -Dx1 , Dx2 , h , -x1*h , -3*x3*Dx3-h^2 ]
! 236: ]
! 237: ]
! 238: Degree shifts
! 239: [ [ 0 ] , [ 0 , 2 , 2 , 2 ] , [ 2 , 2 , 2 , 3 , 3 ] ]
! 240: \end{verbatim}}
! 241: %% $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
! 242: %% $B$O$:(B.
! 243: \end{example}
! 244:
! 245:
! 246:
! 247:
! 248: $(-w,w)$-$B6K>.<+M3J,2r$H(B $B6K>.<+M3J,2r$,$3$H$J$kNc$r$5$,$7$F$$$k$,(B
! 249: $B$3$l$O$^$@8+$D$+$C$F$$$J$$(B.
! 250:
! 251: \section{$B<BAu(B}
! 252: $B$3$3$G$O(B
! 253: \begin{verbatim}
! 254: /* OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v
! 255: 1.23 2000/08/01 08:51:03 takayama Exp */
! 256: \end{verbatim}
! 257: $BHG$N(B {\tt minimal.k} $B$K=`5r$7$F<BAu$N35N,$r2r@b$9$k(B.
! 258:
! 259: $B$^$@=q$$$F$J$$(B.
! 260:
! 261: \end{document}
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