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Annotation of OpenXM/src/k097/lib/minimal/example-ja.tex, Revision 1.3

1.3     ! takayama    1: % $OpenXM: OpenXM/src/k097/lib/minimal/example-ja.tex,v 1.2 2000/08/02 05:14:30 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
                    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}}
1.3     ! takayama  265: %%Prog:test23() of minimal-test.k
        !           266: $B$3$N6K>.<+M3J,2r$O<B$O(B $B9TNs(B $(1,2,3)$ $B$G$-$^$k(B affine toric ideal
        !           267: $B$N6K>.<+M3J,2r$N(B Koszul complex $B$K$J$C$F$k(B.
        !           268: Gel'fand, Kapranov, Zelevinsky $B$K$h$C$FF3F~$5$l$?(B $D/I$
        !           269: $B$N(B resolution $B$r<+A3$K1dD9$7$?<!$N(B 2 $B=EJ#BN$r9M$($h$&(B.
        !           270: $$
        !           271: \begin{array}{ccccccccc}
        !           272: 0 & \longrightarrow & D^2 & \stackrel{d^1}{\longrightarrow}
        !           273:                     & D^3 & \stackrel{d^2}{\longrightarrow}
        !           274:                     & D & \longrightarrow & 0 \\
        !           275:   &                 & u^1 \downarrow      &
        !           276:                     & u^2 \downarrow      &
        !           277:                     & u^3 \downarrow      &   \\
        !           278: 0 & \longrightarrow & D^2 & \stackrel{d^1}{\longrightarrow}
        !           279:                     & D^3 & \stackrel{d^2}{\longrightarrow}
        !           280:                     & D & \longrightarrow & 0
        !           281: \end{array}
        !           282: $$
        !           283: $B$3$3$G$O(B, ($B8m2r$b$J$$$H;W$&$N$G(B) $D$ $B$GF1<!2=%o%$%kBe?t(B,
        !           284: $d^i$ $B$G(B affine toric ideal $B$NF1<!2=$NB?9`<04D$G$N6K>.<+M3J,2r(B
        !           285: $$ d^2 = \pmatrix{ \pd{1}^2 - \pd{2}^2 h \cr
        !           286:                    -\pd{1} \pd{2} + \pd{3} h \cr
        !           287:                    \pd{2}^2 - \pd{1} \pd{3} \cr }, \
        !           288:    d^1 = \pmatrix{ -\pd{2} & -\pd{1} & -h \cr
        !           289:                    \pd{3}  & \pd{2}  & \pd{1} \cr }
        !           290: $$
        !           291: $B$r$"$i$o$9$b$N$H$9$k(B.
        !           292: $B$^$?(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
        !           293: $B<!$N$h$&$K$-$a$k(B.
        !           294: $$ u^1=\pmatrix{ \ell + 4 h^2 & 0 \cr
        !           295:                  0 & \ell+5 h^2 \cr}, \quad
        !           296:    u^2=\pmatrix{\ell+2 h^2 & 0 & 0 \cr
        !           297:                 0 & \ell + 3 h^2 & 0 \cr
        !           298:                 0 & 0 & \ell+ 4 h^2 \cr}, \quad
        !           299:    u^3 = \pmatrix{ \ell \cr}.
        !           300: $$
        !           301:
        !           302: $B$3$N$H$-IU?o$9$k(B 1 $B=EJ#BN$O(B
        !           303: $$L^1 \ni f \mapsto (-d^1(f), u^1(f)) \in L^2 \oplus L^1, $$
        !           304: $$  L^2\oplus L^1 \ni (f,g)\mapsto (-d^2(f), u^2(f)+d^1(g)) \in L^3\oplus L^2,
        !           305: $$
        !           306: $$
        !           307:   L^3\oplus L^2 \ni (f,g)\mapsto u^3(f)+d^2(g) \in L^3.
        !           308: $$
        !           309: $B$G$"$?$($i$l$k(B.
        !           310: $B$3$3$G(B $L^1 = D^2$, $L^2 = D^3$, $L^3 = D$ $B$G$"$k(B.
        !           311: $B$3$N(B 1 $B=EJ#BN$N6qBN7A$O0J2<$N$H$&$j(B.
        !           312: \footnotesize{
        !           313: \begin{verbatim}
        !           314:  [
        !           315:   [
        !           316:     [    x1*Dx1+2*x2*Dx2+3*x3*Dx3 ]
        !           317:     [    Dx1^2-Dx2*h ]
        !           318:     [    -Dx1*Dx2+Dx3*h ]
        !           319:     [    Dx2^2-Dx1*Dx3 ]
        !           320:   ]
        !           321:   [
        !           322:     [    -Dx1^2+Dx2*h , x1*Dx1+2*x2*Dx2+3*x3*Dx3+2*h^2 , 0 , 0 ]
        !           323:     [    Dx1*Dx2-Dx3*h , 0 , x1*Dx1+2*x2*Dx2+3*x3*Dx3+3*h^2 , 0 ]
        !           324:     [    -Dx2^2+Dx1*Dx3 , 0 , 0 , x1*Dx1+2*x2*Dx2+3*x3*Dx3+4*h^2 ]
        !           325:     [    0 , -Dx2 , -Dx1 , -h ]
        !           326:     [    0 , Dx3 , Dx2 , Dx1 ]
        !           327:   ]
        !           328:   [
        !           329:     [    Dx2 , Dx1 , h , x1*Dx1+2*x2*Dx2+3*x3*Dx3+4*h^2 , 0 ]
        !           330:     [    -Dx3 , -Dx2 , -Dx1 , 0 , x1*Dx1+2*x2*Dx2+3*x3*Dx3+5*h^2 ]
        !           331:   ]
        !           332:  ]
        !           333: \end{verbatim}
        !           334: }
1.1       takayama  335: \end{example}
                    336:
                    337:
                    338:
                    339:
1.3     ! takayama  340: \section{$B<BAu(B}
        !           341: $B$3$3$G$O(B
        !           342: \begin{verbatim}
        !           343: /* OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v 1.25
        !           344:    2000/08/02 05:14:31 takayama Exp  */
        !           345: \end{verbatim}
        !           346: $BHG$N(B {\tt minimal.k} $B$K=`5r$7$F<BAu$N35N,$r2r@b$9$k(B.
        !           347:
        !           348: $B<BAu$N@bL@$N$?$a$NNc$H$7$F%$%G%"%k(B
        !           349: $$ I = D \cdot \{  -2x\pd{x}-3y\pd{y}+h^2,  -3y\pd{x}^2+2x\pd{y}h \} $$
        !           350: $B$N(B $(u,v) = (-1,-1,1,1)$-$B6K>.J,2r$N9=@.(B
        !           351: $B$r9M$($h$&(B.
        !           352: %%Prog: minimal-note-ja.txt  6/9 (Fri) $B$*$h$S0J8e$N(B bug fix $B$N5-O?$r;2>H(B.
        !           353: %%$BNc$H$7$F(B, $B%$%G%"%k(B
        !           354: %%$$ I = D \cdot \{ x^2 + y^2, x y \} $$
        !           355: %%$B$N(B $(u,v) = (-1,-1,1,1)$-$B6K>.J,2r$N9=@.(B
        !           356: %%$B$r9M$($h$&(B.
        !           357: %%($B$3$N>l9g$OB?9`<04D$NF1<!<0$G@8@.$5$l$k$N$G(B, $BB?9`<04D$G$N(B
        !           358: %% $B6K>.<+M3J,2r$N7W;;$HF1$8$3$H$K$J$k(B.)
        !           359: $B$3$N>l9g(B,
        !           360: $I$ $B$N%0%l%V%J4pDl(B $G$ $B$O(B
        !           361: {\footnotesize
        !           362: \begin{verbatim}
        !           363:  [
        !           364:    [    -2*x*Dx-3*y*Dy+h^2 ]
        !           365:    [    -3*y*Dx^2+2*x*Dy*h ]
        !           366:    [    9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
        !           367:    [    27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]
        !           368:  ]
        !           369: \end{verbatim}
        !           370: }  \noindent
        !           371: $B$H$J$C$F$*$j(B,
        !           372: Schreyer resolution $B$O(B
        !           373: {\footnotesize
        !           374: \begin{verbatim}
        !           375:   [
        !           376:    [
        !           377:      [    -2*x*Dx-3*y*Dy+h^2 ]
        !           378:      [    -3*y*Dx^2+2*x*Dy*h ]
        !           379:      [    9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
        !           380:      [    27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]
        !           381:    ]
        !           382:    [
        !           383:      [    9*y^2*Dy+3*y*h^2 , 0 , 2*x , 1 ]
        !           384:      [    -4*x^2*Dy*h , 0 , -3*y*Dy+4*h^2 , Dx ]
        !           385:      [    2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]
        !           386:      [    3*y*Dx , -2*x , 1 , 0 ]
        !           387:    ]
        !           388:    [
        !           389:      [    -Dx , 1 , 2*x , 3*y*Dy-2*h^2 ]
        !           390:    ]
        !           391:   ]
        !           392: \end{verbatim}
        !           393: }  \noindent
        !           394: $B$G$"$k(B.  $1$ $B$,$?$/$5$s(B Schreyer resolution $B$NCf$K$O$"$k$3$H$K(B
        !           395: $BCm0U(B. $1$ $B$O6K>.<+M3J,2r$K$OI,MW$J$$85$G$"$k$3$H$r0UL#$9$k(B.
        !           396: $B6K>.<+M3J,2r$O(B, $BNc(B \ref{example:cusp} $B$K6qBN7A$r=q$$$F$*$$$?(B.
        !           397:
        !           398: \medbreak
        !           399:
        !           400:
        !           401: $B$3$N<BAu$G$O6K>.<+M3J,2r$r(B LaScala $B$N%"%k%4%j%:%`$r$b$H$K$7$F(B
        !           402: $B9=@.$9$k(B (LaScala and Stillman [??] $B$*$h$S?tM}2J3X$N5-;v(B ??? $B$r;2>H(B).
        !           403:
        !           404: $B$3$N%"%k%4%j%:%`$O4{CN$H$7$F(B, $B0c$$$N$_$r@bL@$7$h$&(B.
        !           405: LaScala $B$N%"%k%4%j%:%`$O(B,
        !           406: reduction $B$7$?$H$-$K(B $0$ $B$K$J$C$?>l9g(B, $B$=$N(B reduction $B$KIU?o$7$?(B
        !           407: syzygy $B$r(B $B6K>.<+M3J,2r$N85$H$7(B,
        !           408: reduction $B$7$?$H$-$K(B $0$ $B$K$J$i$J$+$C$?>l9g(B, $B$=$N85$r(B
        !           409: $B%0%l%V%J4pDl$N85$H$7$F2C$((B, $BIU?o$7$?(B syzygy $B$O6K>.<+M3J,2r$G$OM>7W$J$b$N$H(B
        !           410: $B$_$J$9(B.
        !           411: $B$o$l$o$l$O(B $(u,v)$-$B6K>.$J<+M3J,2r$r$b$H$a$?$$(B.
        !           412: $B$=$3$G>e$N<jB3$-$r<!$N$h$&$KJQ$($k(B.
        !           413: \begin{center}
        !           414: \begin{minipage}{10cm}
        !           415: Reduction $B$7$?$H$-$K(B modulo $(u,v)$-$B%U%#%k%?!<$G(B $0$ $B$K$J$C$?>l9g(B,
        !           416: $B$=$N(B reduction $B$KIU?o$7$?(B syzygy $B$r(B $B6K>.<+M3J,2r$N85$H$7(B, $B$5$i$K(B
        !           417: $B$=$N85$,(B $0$ $B$G$J$1$l$P%0%l%V%J4pDl$K2C$($k(B.
        !           418: reduction $B$7$?$H$-$K(B modulo $(u,v)$-$B%U%#%k%?!<$G(B $0$ $B$K$J$i$J$+$C$?>l9g(B,
        !           419: $B$=$N85$r%0%l%V%J4pDl$N85$H$7$F2C$((B,
        !           420: $BIU?o$7$?(B syzygy $B$O6K>.<+M3J,2r$G$OM>7W$J$b$N$H$_$J$9(B.
        !           421: \end{minipage}
        !           422: \end{center}
        !           423:
        !           424: $B$A$J$_$K(B,
1.1       takayama  425: $(-w,w)$-$B6K>.<+M3J,2r$H(B $B6K>.<+M3J,2r$,$3$H$J$kNc$r$5$,$7$F$$$k$,(B
                    426: $B$3$l$O$^$@8+$D$+$C$F$$$J$$(B.
1.3     ! takayama  427: $B$A$g$C$HIT;W5D$G$"$k(B.
        !           428:
        !           429: \bigbreak
        !           430:
        !           431: {\tt minimal.k} $B$N%=!<%9%3!<%I$G$O$3$NItJ,$O<!$N$h$&$K$J$C$F$$$k(B.
        !           432: {\footnotesize
        !           433: \begin{verbatim}
        !           434: def SlaScala(g,opt) {
        !           435:     ...
        !           436:     ...
        !           437:                f = SpairAndReduction(skel,level,i,freeRes,tower,ww);
        !           438:                if (f[0] != Poly("0")) {
        !           439:                   place = f[3];
        !           440: if (Sordinary) {
        !           441:                   redundantTable[level-1,place] = redundant_seq;
        !           442:                   redundant_seq++;
        !           443: }else{
        !           444:                   if (f[4] > f[5]) {                       ($B$$(B)
        !           445:                     /* Zero in the gr-module */
        !           446:                     Print("v-degree of [org,remainder] = ");
        !           447:                     Println([f[4],f[5]]);
        !           448:                     Print("[level,i] = "); Println([level,i]);
        !           449:                     redundantTable[level-1,place] = 0;
        !           450:                   }else{                                   ($B$m(B)
        !           451:                     redundantTable[level-1,place] = redundant_seq;
        !           452:                     redundant_seq++;
        !           453:                   }
        !           454: }
        !           455:                   redundantTable_ordinary[level-1,place]
        !           456:                      =redundant_seq_ordinary;
        !           457:   ...
        !           458:   ...
        !           459: }
        !           460: \end{verbatim}
        !           461: }
        !           462:
        !           463: $B>/!9D9$/$J$k$,(B, $B$3$NItJ,$K$"$i$o$l$kJQ?t$N@bL@$r$7$h$&(B.
        !           464:
        !           465: LaScala $B$N%"%k%4%j%:%`$G$O(B, $B:G=i$K7W;;$9$Y$-(B S-pair $B$N7W;;<j=g(B,
        !           466: $B$*$h$S(B Schreyer frame $B$r:n@.$9$k(B.
        !           467: Schreyer frame $B$O(B Schreyer resolution $B$N(B initial $B$G$"$k(B.
        !           468: $B$3$l$i$O$"$i$+$8$a(B
        !           469: {\tt SresolutionFrameWithTower(g,opt);}
        !           470: $B$G7W;;$5$l$F(B, {\tt tower} $B$*$h$S(B {\tt skel} $B$K3JG<$5$l$F$$$k(B.
        !           471: $B$3$l$i$NJQ?t$NCM$O(B, $B4X?t(B {\tt Sminimal()} $B$NLa$jCM$H$7$F(B
        !           472: $B8+$k$3$H$,$G$-$k(B.
        !           473: $B4X?t(B {\tt Sminimal()} $B$NLa$jCM$,JQ?t(B $a$ $B$K3JG<$5$l$F$$$k$H$9$k$H(B,
        !           474: {\tt a[0]} $B$,6K>.<+M3J,2r(B
        !           475: {\tt a[3]} $B$,(B Schreyer $B<+M3J,2r(B($B$H$/$K(B {\tt a[3,0]} $B$,(B
        !           476: $I$ $B$N%0%l%V%J4pDl(B),
        !           477: {\tt a[4]} $B$,(B,
        !           478: $B4X?t(B {\tt SlaScala()} $B$N(B
        !           479: $BJQ?t(B {\tt [rf[0], tower, skel, rf[3]]} $B$NCM$G$"$k(B.
        !           480: $B$7$?$,$C$F(B,  {\tt tower} $B$O(B {\tt a[4,1]} $B$K3JG<$5$l$F$$$k(B.
        !           481: $I$ $B$N>l9g$N(B {\tt tower} $B$O0J2<$N$H$&$j(B.
        !           482: {\footnotesize
        !           483: \begin{verbatim}
        !           484: In(25)=sm1_pmat(a[4,1]);
        !           485:  [
        !           486:    [    -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ]
        !           487:    [    -9*y^2*Dy , -3*es^2*y*Dy , -3*es*y*Dy , -3*y*Dx ]
        !           488:    [    -Dx ]
        !           489:  ]
        !           490: \end{verbatim}
        !           491: } \noindent
        !           492: $B$3$3$G(B ${\tt es}^i$ $B$O%Y%/%H%k$N(B $BBh(B $i$ $B@.J,$G$"$k$3$H$r$7$a$7$F$$$k(B.
        !           493: $B$?$H$($P(B,
        !           494: \verb# -3*es^2*y*Dy # $B$O(B
        !           495: \verb# [0, 0, -3*y*Dy, 0] # $B$r0UL#$9$k(B.
        !           496:
        !           497: $BJQ?t(B
        !           498: {\tt skel} $B$K$O(B
        !           499: S-pair (sp) $B$N7W;;<j=g$,$O$$$C$F$$$k(B.
        !           500: $I$ $B$N>l9g$K$O0J2<$N$H$&$j(B.
        !           501: {\footnotesize
        !           502: \begin{verbatim}
        !           503: In(16)=sm1_pmat(a[4,2]);
        !           504:  [
        !           505:    [   ]
        !           506:   [
        !           507:    [
        !           508:      [    0 , 2 ]          G'[0] $B$H(B G'[2] $B$N(B sp $B$r7W;;(B (0)
        !           509:      [    -9*y^2*Dy , 2*x ]
        !           510:    ]
        !           511:    [
        !           512:      [    2 , 3 ]          G'[2] $B$H(B G'[3] $B$N(B sp $B$r7W;;(B (1)
        !           513:      [    -3*y*Dy , Dx ]
        !           514:    ]
        !           515:    [
        !           516:      [    1 , 2 ]          G'[1] $B$H(B G'[2] $B$N(B sp $B$r7W;;(B (2)
        !           517:      [    -3*y*Dy , Dx ]
        !           518:    ]
        !           519:    [
        !           520:      [    0 , 1 ]          G'[0] $B$H(B G'[1] $B$N(B sp $B$r7W;;(B (3)
        !           521:      [    -3*y*Dx , 2*x ]
        !           522:    ]
        !           523:   ]
        !           524:   [
        !           525:    [
        !           526:      [    0 , 3 ]          G''[0] $B$H(B G''[3] $B$N(B sp $B$r7W;;(B
        !           527:      [    -Dx , 3*y*Dy ]
        !           528:    ]
        !           529:   ]
        !           530:    [   ]
        !           531:  ]
        !           532: \end{verbatim}
        !           533: }  \noindent
        !           534: $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,
        !           535: $G''$ $B$O(B Schreyer order $B$GF@$i$l$?(B $G'$ $B$N(B syzygy $B$N@8@.85$r$"$i$o$9(B.
        !           536: $B$?$H$($P>e$NNc$G$O(B,
        !           537: $G'[0]$ $B$O(B
        !           538: $G[0]$ $B$H(B $G[2]$ $B$N(B sp $B$N7W;;$K$h$jF@$i$l$?(B syzygy,
        !           539: $G'[1]$ $B$O(B
        !           540: $G[2]$ $B$H(B $G[3]$ $B$N(B sp $B$N7W;;$K$h$jF@$i$l$?(B syzygy,
        !           541: ...
        !           542: $B$r0UL#$9$k(B.
        !           543:
        !           544: {\footnotesize
        !           545: \begin{verbatim}
        !           546:      f = SpairAndReduction(skel,level,i,freeRes,tower,ww);
        !           547: \end{verbatim}
        !           548: } \noindent
        !           549: $B$G$O(B {\tt skel[level,i]} $B$K3JG<$5$l$?(B
        !           550: S-pair $B$r7W;;$7$F(B, {\tt freeRes[level-1]} $B$G(B reduction $B$r$*$3$J$&(B.
        !           551: Reduction $B$N$?$a$N(B Schreyer order $B$O(B \\
        !           552: {\tt StowerOf(tower,level-1)} $B$rMQ$$$k(B.
        !           553: $B$?$H$($P(B, ${\tt [level,i] = [1,3]}$ $B$N$H$-$K(B
        !           554: $B4X?t(B {\tt SpairAndReduction} $B$G(B
        !           555: $B$I$N$h$&$J7W;;$,$J$5$l$F$$$k$+(B $I$ $B$N>l9g$K$_$F$_$h$&(B.
        !           556:
        !           557: {\tt SpairAndReduction} $B$N<B9T;~(B
        !           558: $B$K<!$N$h$&$J%a%C%;!<%8$,$G$F$/$k(B.
        !           559: {\footnotesize
        !           560: \begin{verbatim}
        !           561: reductionTable= [
        !           562:    [    1 , 2 , 3 , 4 ]
        !           563:    [    3 , 4 , 3 , 2 ]
        !           564:    [    3 ]
        !           565:  ]
        !           566: [    0 , 0 ]
        !           567: Processing [level,i]= [    0 , 0 ]    Strategy = 1
        !           568: [    0 , 1 ]
        !           569: Processing [level,i]= [    0 , 1 ]    Strategy = 2
        !           570: [    1 , 3 ]
        !           571: Processing [level,i]= [    1 , 3 ]    Strategy = 2
        !           572: SpairAndReduction:
        !           573: [    p and bases  , [    [    0 , 1 ]  , [    -3*y*Dx , 2*x ]  ]  ,
        !           574:    [-2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ]  ]
        !           575: [    level= , 1 ]
        !           576: [    tower2= , [    [   ]  ]  ]
        !           577: [    -3*y*Dx , 2*es*x ]
        !           578: [gi, gj] = [    -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h ]
        !           579: 1
        !           580: Reduce the element 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h
        !           581: by  [ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ]
        !           582: result is [    9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , 1 , [    0 , 0 , 0 , 0 ]  ]
        !           583: vdegree of the original = 0
        !           584: vdegree of the remainder = 0
        !           585: [  9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ,
        !           586:   [ -3*y*Dx , 2*x , 0 , 0 ]  , 3 , 2 , 0 , 0 ]
        !           587: \end{verbatim}
        !           588: }  \noindent
        !           589: $B:G=i$KI=<($5$l$k(B {\tt reductionTable} $B$N0UL#$O$"$H$G@bL@$9$k(B.
        !           590: $B<!$N9T$KCmL\$7$h$&(B.  $B$3$3$G$O(B {\tt skel[0,4]} $B$N(B S-pair
        !           591: $B$r7W;;$7$F(Breduction $B$7$F$$$k(B.
        !           592: {\footnotesize
        !           593: \begin{verbatim}
        !           594: SpairAndReduction:
        !           595: [    p and bases  , [    [    0 , 1 ]  , [    -3*y*Dx , 2*x ]  ]  ,
        !           596:    [-2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ]  ]
        !           597: \end{verbatim}
        !           598: }  \noindent
        !           599: {\tt [0, 1]} $B$O(B  $G'[0]$ $B$H(B $G'[1]$ $B$N(B sp $B$r7W;;(B
        !           600: $B$;$h$H$$$&0UL#$G$"$k(B.
        !           601: ${\tt level} = 0$ $B$G4{$K$b$H$^$C$F$$$k(B $B%V%l%V%J4pDl$O(B
        !           602: $G[0]$ $B$H(B $G[1]$ $B$N$_$G$"$j(B,
        !           603: $B$=$l$i$O$=$l$>$l(B,
        !           604: \verb# -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h #
        !           605: $B$G$"$k(B.
        !           606: {\tt SpairAndReduction} $B$O(B $G[0]$, $G[1]$ $B$N$_$rMQ$$$F(B,
        !           607: S-pair  \\
        !           608: \verb# 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h #
        !           609: $B$r(B reduction $B$9$k(B.
        !           610: $B7k6I(B reduction $B$N7k2L$O(B 0 $B$G$O$J$/$F(B, \\
        !           611: \verb# 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h #
        !           612: $B$H$J$k(B.
        !           613: LaScala $B$N%"%k%4%j%:%`$N(B 2 $BG\$*F@%7%9%F%`$G(B,
        !           614: $B$3$l$,?7$7$$%0%l%V%J4pDl$N85(B {\tt G[place]} $B$H$J$j(B,
        !           615: reduction $B$N2aDx$h$j(B syzygy $B$bF@$i$l$k(B.
        !           616:
        !           617: $B$5$F(B, $(u,v)$-$B6K>.J,2r$r:n$k$K$O(B, reduction $B$7$?M>$j$,(B
        !           618: $(u,v)$-$B%U%#%k%?!<$G(B modulo $B$7$F(B $0$ $B$+$I$&$+D4$Y$J$$$H$$$1$J$$(B.
        !           619: $B$3$N$?$a(B,
        !           620: $B4X?t(B {\tt Sdegree()} $B$rMQ$$$F(B, reduction $B$9$kA0$N85(B, $B$*$h$SM>$j$N(B
        !           621: $B%7%U%HIU$-(B $(u,v)$-order $B$r7W;;$9$k(B.
        !           622: $B$3$NNc$G$O(B, $BN>J}$H$b(B $0$ $B$G$"$k(B.
        !           623: {\footnotesize
        !           624: \begin{verbatim}
        !           625: vdegree of the original = 0
        !           626: vdegree of the remainder = 0
        !           627: \end{verbatim}
        !           628: }
        !           629: $B$7$?$,$C$F(B, modulo $(u,v)$-$B%U%#%k%?!<$G$b(B $0$ $B$G$J$$(B.
1.1       takayama  630:
1.3     ! takayama  631: $B=`Hw@bL@$,$*$o$C$?(B. $B:G=i$N%W%m%0%i%`(B {\tt SlaScala()} $B$N@bL@$KLa$k(B.
        !           632: {\tt SpairAndReduction()} $B$NLa$jCM(B
        !           633: {\tt f[0]} $B$K$O(B, reduction $B$7$?M>$j(B,
        !           634: {\tt f[4]}, {\tt f[5]} $B$K$O(B,
        !           635: reduction $B$9$kA0$N85(B, $B$*$h$SM>$j$N(B
        !           636: $B%7%U%HIU$-(B $(u,v)$-order $B$,3JG<$5$l$F$$$k(B.
        !           637: $B$3$NNc$N>l9g$K$O(B ($B$m(B) $B$N>l9g$,<B9T$5$l$F(B,
        !           638: $BIU?o$7$?(B syzygy $B$O(B $B6K>.<+M3J,2r$K$OITMW$J$b$N$H$7$F(B
        !           639: {\tt redundantTable} $B$KEPO?$5$l$k(B:
        !           640: {\footnotesize
        !           641: \begin{verbatim}
        !           642:                     redundantTable[level-1,place] = redundant_seq;
        !           643: \end{verbatim}
        !           644: }  \noindent
        !           645: $BM>$j(B {\tt f[0]} $B$O(B, laScala $B$N%"%k%4%j%:%`$N(B 2 $BG\$*F@8=>]$GF@$i$l$?(B,
        !           646: $B?7$7$$%V%l%V%J4pDl$N85$G$"$k$,(B, $B$3$l$rJ]B8$9$Y$->l=j$N%$%s%G%C%/%9$O(B,
        !           647: $BLa$jCM(B {\tt f[3]}({\tt place}) $B$K3JG<$5$l$F$$$k(B:
        !           648: {\footnotesize
        !           649: \begin{verbatim}
        !           650:                   bases[place] = f[0];
        !           651:                   freeRes[level-1] = bases;
        !           652:                   reducer[level-1,place] = f[1];
        !           653: \end{verbatim}
        !           654: } \noindent
        !           655: $B$3$N(B reduction $B$GF@$i$l$?(B syzygy ($B$NK\<AE*ItJ,(B)$B$O(B,
        !           656: $BJQ?t(B {\tt reducer} $B$KEPO?$5$l$k(B.
        !           657: $B0J>e$G(B $(u,v)$-$B6K>.<+M3J,2rFCM-$N=hM}$NItJ,$N2r@b$r=*$($k(B.
        !           658:
        !           659:
        !           660: \bigbreak
        !           661: $B0J2<$G$O(B, LaScala $B$N%"%k%4%j%:%`$N$o$l$o$l$N<BAu$N35N,$HLdBjE@$r(B
        !           662: $B=R$Y$k(B.
        !           663:
        !           664: $B$^$:(B, $BJQ?t(B
        !           665: {\tt reductionTable} $B$N0UL#$r@bL@$7$h$&(B.
        !           666: LaScala $B$N%"%k%4%j%:%`$G$O(B,
        !           667: {\tt level - Sdegree(s)}
        !           668: $B$N>.$5$$(B S-pair $B$+$i7W;;$7$F$$$/(B.
        !           669: $B4X?t(B {\tt Sdegree} $B$O<!$N$h$&$K:F5"E*$KDj5A$5$l$F$$$k(B.
        !           670: {\footnotesize
        !           671: \begin{verbatim}
        !           672: /* f is assumed to be a monomial with toes. */
        !           673: def Sdegree(f,tower,level) {
        !           674:   local i,ww, wd;
        !           675:   /* extern WeightOfSweyl; */
        !           676:   ww = WeightOfSweyl;
        !           677:   f = Init(f);
        !           678:   if (level <= 1) return(StotalDegree(f));
        !           679:   i = Degree(f,es);
        !           680:   return(StotalDegree(f)+Sdegree(tower[level-2,i],tower,level-1));
        !           681: }
        !           682: \end{verbatim}
        !           683: }  \noindent
        !           684: $B$3$3$G(B {\tt StotalDegree(f)} $B$O(B $f$ $B$NA4<!?t$G$"$k(B.
        !           685:
        !           686: \noindent
        !           687: $B$5$F(B, LaScala $B$N%"%k%4%j%:%`$G$O(B,
        !           688: Resolution $B$r2<$+$i=gHV$K7W;;$7$F$$$/$N$G$O$J$$(B.
        !           689: $B$3$l$,K\<AE*$JE@$G$"$k(B.
        !           690: $B$3$N=gHV$OJQ?t(B {\tt reductionTable} $B$K$O$C$F$$$k(B.
        !           691: $I$ $B$NNc$G$O(B
        !           692: {\footnotesize
        !           693: \begin{verbatim}
        !           694: reductionTable= [
        !           695:    [    1 , 2 , 3 , 4 ]
        !           696:    [    3 , 4 , 3 , 2 ]   skel[0] $B$KBP1~(B
        !           697:    [    3 ]               skel[1] $B$KBP1~(B
        !           698:  ]
        !           699: \end{verbatim}
        !           700: }  \noindent
        !           701: $B$H$J$k(B.
        !           702:
        !           703: $B8=:_$N<BAu$G$N7W;;B.EY(B, $B%a%b%j;HMQNL$N%\%H%k%M%C%/$r(B
        !           704: $B;XE&$7$F$*$/(B.
        !           705: LaScala $B$N%"%k%4%j%:%`$G$O(B, Schreyer Frame $B$r9=@.$7$F$+$i(B,
        !           706: $B6K>.<+M3J,2r$r9=@.$9$k(B.
        !           707: $B2<5-$N%W%m%0%i%`$NJQ?t(B {\tt redundantTable[level,q]} $B$K$O(B,
        !           708: $BBP1~$9$k(B syzygy $B$H(B $B%0%l%V%J4pDl$N85$,2?2sL\$N(B reduction $B$G@8@.(B
        !           709: $B$5$l$?$+$N?t$,$O$$$C$F$$$k(B.
        !           710: $B6K>.<+M3J,2r$N9=@.$G$O(B, $B:G8e$N(B reduction $B$N(B syzygy $B$+$i;O$a$F(B,
        !           711: Schreyer resolution $B$+$i6K>.<+M3J,2r$K$H$C$FM>J,$J85$r<h$j=|$$$F(B
        !           712: $B$$$/(B
        !           713: ({\tt seq} $B$r(B $1$ $B$E$D8:$i$7$F$$$/(B).
        !           714: {\footnotesize
1.1       takayama  715: \begin{verbatim}
1.3     ! takayama  716: def Sminimal(g,opt) {
        !           717:
        !           718:   ....
        !           719:
        !           720:   while (seq > 1) {
        !           721:     seq--;
        !           722:     for (level = 0; level < maxLevel; level++) {
        !           723:       betti = Length(freeRes[level]);
        !           724:       for (q = 0; q<betti; q++) {
        !           725:         if (redundantTable[level,q] == seq) {
        !           726:           Print("[seq,level,q]="); Println([seq,level,q]);
        !           727:           if (level < maxLevel-1) {
        !           728:             bases = freeRes[level+1];
        !           729:             dr = reducer[level,q];
        !           730:             dr[q] = -1;
        !           731:             newbases = SnewArrayOfFormat(bases);
        !           732:             betti_levelplus = Length(bases);
        !           733:             /*
        !           734:                bases[i,j] ---> bases[i,j]+bases[i,q]*dr[j]
        !           735:             */
        !           736:             for (i=0; i<betti_levelplus; i++) {
        !           737:               newbases[i] = bases[i] + bases[i,q]*dr;
        !           738:             }
        !           739:             ....
        !           740:           }
        !           741:           ....
        !           742:         }
        !           743:      }
        !           744:    }
        !           745:   }
        !           746:   ....
        !           747: }
1.1       takayama  748: \end{verbatim}
1.3     ! takayama  749: } \noindent
        !           750: $BLdBj$O(B,
        !           751: $B6K>.<+M3J,2r<+BN$O$A$$$5$/$F$b(B, Schreyer Frame $B$,5pBg(B ($10000$ $BDxEY$N(B
        !           752: betti $B?t(B) $B$H$J$k$3$H$bB?$$>l9g$,$"$k$3$H$G$"$k(B.
        !           753: $B2<$NJQ?t(B {\tt bases} $B$K(B, Schreyer resolution $B$N(B {\tt level} $B<!$N(B
        !           754: syzygy $B$r$$$l$F$$$k(B. Schreyer Frame $B$K(B $10000$ $BDxEY$N(B betti
        !           755: $B?t$,$"$i$o$l$k$H$3$NJQ?t$O(B $B%5%$%:(B $10000$ $BDxEY$NG[Ns$H$J$k(B.
        !           756: $B$5$i$K(B, Schreyer $BJ,2r$+$i6K>.<+M3J,2r$N$?$a$KITMW$J85$r$H$j$N$>$$$?(B
        !           757: $BJ,2r$r:n$k$?$a$K(B\\
        !           758: \verb#              newbases[i] = bases[i] + bases[i,q]*dr;   # \\
        !           759: $B$J$k>C5n$r$*$3$J$$(B, $0$ $B$GKd$a$i$l$?Ns$^$?$O(B $0$ $B$GKd$a$i$l$?9T$r@8@.$7$F$$$k(B.
        !           760: $B$3$NItJ,$,(B, $B%a%b%j$N;HMQ$r05Gw$7$F$*$j(B, $B7W;;;~4V$b$D$+$C$F$$$k(B.
        !           761:
1.1       takayama  762:
                    763:
                    764: \end{document}