| version 1.1, 2000/08/02 03:23:36 |
version 1.3, 2000/08/09 03:45:27 |
|
|
| % $OpenXM$ |
% $OpenXM: OpenXM/src/k097/lib/minimal/example-ja.tex,v 1.2 2000/08/02 05:14:30 takayama Exp $ |
| \documentclass[12pt]{jarticle} |
\documentclass[12pt]{jarticle} |
| \newtheorem{example}{Example} |
\newtheorem{example}{Example} |
| \def\pd#1{ \partial_{#1} } |
\def\pd#1{ \partial_{#1} } |
| Line 39 tie-breaking order $B$K$b0MB8$9$k(B. |
|
| Line 39 tie-breaking order $B$K$b0MB8$9$k(B. |
|
| \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 |
\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 |
| ${\rm Ann}(D f^{-1})$ $B$G(B |
${\rm Ann}(D f^{-1})$ $B$G(B |
| $1/f^r$ $B$rNm2=$9$k(B $D$ $B$N%$%G%"%k$N$"$k@8@.85$N=89g$r$"$i$o$9(B. |
$1/f^r$ $B$rNm2=$9$k(B $D$ $B$N%$%G%"%k$N$"$k@8@.85$N=89g$r$"$i$o$9(B. |
| $B2<$N<BNc$G$O4X?t(B {\tt Sannfs(f,v)} $B$N=PNO$r$"$i$o$9(B. |
$B2<$N<BNc$N>l9g$G$O4X?t(B {\tt Sannfs(f,v)} $B$N=PNO$r$"$i$o$9(B. |
| \item $F(G)$ $B$G(B $G$ $B$N(B formal Laplace $BJQ49$r$"$i$o$9(B. |
\item $F(G)$ $B$G(B $G$ $B$N(B formal Laplace $BJQ49$r$"$i$o$9(B. |
| \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. |
\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. |
| \item Grothendieck $B$NHf3SDjM}$K$h$l$P(B |
\item Grothendieck $B$NHf3SDjM}$K$h$l$P(B |
| Line 50 complement of an affine variety via D-module computati |
|
| Line 50 complement of an affine variety via D-module computati |
|
| Journal of pure and applied algebra, 139 (1999), 201--233. $B$r;2>H(B) |
Journal of pure and applied algebra, 139 (1999), 201--233. $B$r;2>H(B) |
| \end{enumerate} |
\end{enumerate} |
| |
|
| \begin{example} \rm |
\begin{example} \rm \label{example:cusp} |
| %Prog: minimal-test.k test18() |
%Prog: minimal-test.k test18() |
| $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2} \right) \right]$ |
$I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2} \right) \right]$ |
| $B$N>l9g(B. |
$B$N>l9g(B. |
| Line 61 $$ -2x\pd{x}-3y\pd{y}+h^2 , -3y\pd{x}^2+2x\pd{y}h $$ |
|
| Line 61 $$ -2x\pd{x}-3y\pd{y}+h^2 , -3y\pd{x}^2+2x\pd{y}h $$ |
|
| \begin{tabular}{|l|l|} |
\begin{tabular}{|l|l|} |
| \hline |
\hline |
| Resolution type & Betti numbers \\ \hline |
Resolution type & Betti numbers \\ \hline |
| Schreyer & 2, 1 \\ \hline |
Schreyer & 1, 4, 4, 1 \\ \hline |
| $(-{\bf 1},{\bf 1})$-minimal & 4, 4, 1 \\ \hline |
$(-{\bf 1},{\bf 1})$-minimal & 1, 2, 1 \\ \hline |
| minimal & 2, 1 \\ |
minimal & 1, 2, 1 \\ |
| \hline |
\hline |
| \end{tabular} |
\end{tabular} |
| |
|
| Line 112 $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2z^2} \ |
|
| Line 112 $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2z^2} \ |
|
| \begin{tabular}{|l|l|} |
\begin{tabular}{|l|l|} |
| \hline |
\hline |
| Resolution type & Betti numbers \\ \hline |
Resolution type & Betti numbers \\ \hline |
| Schreyer & 4, 5, 2 \\ \hline |
Schreyer & 1, 8, 16, 11, 2 \\ \hline |
| $(-{\bf 1},{\bf 1})$-minimal & 8, 16, 11, 2 \\ \hline |
$(-{\bf 1},{\bf 1})$-minimal & 1, 4, 5, 2 \\ \hline |
| minimal & 4, 5, 2 \\ |
minimal & 1, 4, 5, 2 \\ |
| \hline |
\hline |
| \end{tabular} |
\end{tabular} |
| |
|
| Line 154 $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3+y^3+z^3} |
|
| Line 154 $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3+y^3+z^3} |
|
| \begin{tabular}{|l|l|} |
\begin{tabular}{|l|l|} |
| \hline |
\hline |
| Resolution type & Betti numbers \\ \hline |
Resolution type & Betti numbers \\ \hline |
| Schreyer & \\ \hline |
Schreyer & 1, 12, 44, 75, 70, 39, 13, 2 \\ \hline |
| $(-{\bf 1},{\bf 1})$-minimal & \\ \hline |
$(-1,-2,-3,1,2,3)$-minimal & 1, 4, 5, 2 \\ \hline |
| minimal & \\ |
minimal & 1, 4, 5, 2 \\ |
| \hline |
\hline |
| \end{tabular} |
\end{tabular} |
| |
|
| \noindent |
\noindent |
| $(-{\bf 1},{\bf 1})$-minimal resolution |
$(-1,-2,-3,1,2,3)$-minimal resolution |
| {\footnotesize \begin{verbatim} |
{\footnotesize \begin{verbatim} |
| |
[ |
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[ |
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[ x*Dx+y*Dy+z*Dz-3*h^2 ] |
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[ y*Dz^2-z*Dy^2 ] |
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[ x*Dz^2-z*Dx^2 ] |
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[ x*Dy^2-y*Dx^2 ] |
| |
] |
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[ |
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[ 0 , -x , y , -z ] |
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[ -x*Dz^2+z*Dx^2 , x*Dy , x*Dx+z*Dz-3*h^2 , z*Dy ] |
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[ -x*Dy^2+y*Dx^2 , -x*Dz , y*Dz , x*Dx+y*Dy-3*h^2 ] |
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[ -y*Dz^2+z*Dy^2 , x*Dx+y*Dy+z*Dz-2*h^2 , 0 , 0 ] |
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[ 0 , Dx^2 , -Dy^2 , Dz^2 ] |
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] |
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[ |
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[ -x*Dx+3*h^2 , y , -z , -x , 0 ] |
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[ -Dz^3-Dy^3 , -Dy^2 , Dz^2 , Dx^2 , -x*Dx-y*Dy-z*Dz ] |
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] |
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] |
| |
Degree shifts |
| |
[ [ 0 ] , [ 0 , 4 , 5 , 3 ] , [ 3 , 5 , 6 , 4 , 9 ] ] |
| \end{verbatim}} |
\end{verbatim}} |
| \end{example} |
\end{example} |
| |
|
| Line 176 $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2z^2+y^ |
|
| Line 196 $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2z^2+y^ |
|
| \begin{tabular}{|l|l|} |
\begin{tabular}{|l|l|} |
| \hline |
\hline |
| Resolution type & Betti numbers \\ \hline |
Resolution type & Betti numbers \\ \hline |
| Schreyer & \\ \hline |
Schreyer & 1, 13, 43, 50, 21, 2 \\ \hline |
| $(-{\bf 1},{\bf 1})$-minimal & \\ \hline |
$(-{\bf 1},{\bf 1})$-minimal & 1, 7, 10, 4 \\ \hline |
| minimal & \\ |
minimal & 1, 7, 10, 4 \\ |
| \hline |
\hline |
| \end{tabular} |
\end{tabular} |
| |
|
| \noindent |
\noindent |
| $(-{\bf 1},{\bf 1})$-minimal resolution |
$f=x^3-y^2z^2+y^2+z^2$ $B$H$*$$$?>l9g(B, |
| {\footnotesize \begin{verbatim} |
$B6u4V(B ${\bf C}^3 \setminus V(f)$ $B$N(B |
| |
$B%3%[%b%m%872$N<!85$O(B |
| \end{verbatim}} |
${\rm dim}\, H^i = 1$, $(i=0, 1)$, |
| $B%3%[%b%m%872$O(B ... $B$H$J$k(B. |
${\rm dim}\, H^i = 0$, $(i=2, 3)$, |
| $B9M$($k@~7A6u4V$NJ#BN$N<!85$O(B, ... |
$B$H$J$k(B. |
| Schreyer resolution $B$+$i%9%?!<%H$7$F(B, |
$B$3$N>l9g(B $D/I$ $B$N(B |
| |
$b$-$B4X?t$N:GBg@0?t:,$O(B $2$ $B$H$J$j(B, |
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$B%3%[%b%m%8$r7W;;$9$k$?$a$K(B |
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$B9M$($k@~7A6u4V$NJ#BN$N<!85$O(B, $10, 12, 9, 4$ $B$G$"$k(B. %%Prog: Srestall.sm1 |
| |
$B0lJ}(B Schreyer resolution $B$+$i%9%?!<%H$7$F(B, |
| $B@~7A6u4V$NJ#BN$r9M$($k$H(B, $B$=$N<!85$O(B |
$B@~7A6u4V$NJ#BN$r9M$($k$H(B, $B$=$N<!85$O(B |
| ... $B$H$J$k(B. |
130, 1078, 1667, 749, 40 $B$H$J$k(B. %%Prog: test21b() |
| \end{example} |
\end{example} |
| |
|
| \begin{example} \rm |
\begin{example} \rm |
| %Prog: minimal-test.k test20() |
%Prog: minimal-test.k test20() |
| $I = D\cdot\{ x_1*\pd{1}+2x_2\pd{2}+3x_3\pd{3} , |
$I = D\cdot\{ x_1\pd{1}+2x_2\pd{2}+3x_3\pd{3} , |
| \pd{1}^2-\pd{2}*h, |
\pd{1}^2-\pd{2}h, |
| -\pd{1}\pd{2}+\pd{3}*h, |
-\pd{1}\pd{2}+\pd{3}h, |
| \pd{2}^2-\pd{1}\pd{3} \} |
\pd{2}^2-\pd{1}\pd{3} \} |
| $ $B$N>l9g(B. |
$ $B$N>l9g(B. |
| $B$3$l$O(B $A=(1,2,3)$, $\beta=0$ $B$KIU?o$9$k(B GKZ $BD64v2?7O$N(B |
$B$3$l$O(B $A=(1,2,3)$, $\beta=0$ $B$KIU?o$9$k(B GKZ $BD64v2?7O$N(B |
|
|
| \begin{tabular}{|l|l|} |
\begin{tabular}{|l|l|} |
| \hline |
\hline |
| Resolution type & Betti numbers \\ \hline |
Resolution type & Betti numbers \\ \hline |
| Schreyer & 4, 5, 2 \\ \hline |
Schreyer & 1, 10, 25, 23, 8, 1 \\ \hline |
| $(-{\bf 1},{\bf 1})$-minimal & 10, 25, 23, 8, 1 \\ \hline |
$(-{\bf 1},{\bf 1})$-minimal & 1, 4, 5, 2 \\ \hline |
| minimal & 4, 5, 2 \\ |
minimal & 1, 4, 5, 2 \\ |
| \hline |
\hline |
| \end{tabular} |
\end{tabular} |
| |
|
| Line 238 $(-{\bf 1},{\bf 1})$-minimal resolution |
|
| Line 262 $(-{\bf 1},{\bf 1})$-minimal resolution |
|
| Degree shifts |
Degree shifts |
| [ [ 0 ] , [ 0 , 2 , 2 , 2 ] , [ 2 , 2 , 2 , 3 , 3 ] ] |
[ [ 0 ] , [ 0 , 2 , 2 , 2 ] , [ 2 , 2 , 2 , 3 , 3 ] ] |
| \end{verbatim}} |
\end{verbatim}} |
| %% $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 |
%%Prog:test23() of minimal-test.k |
| %% $B$O$:(B. |
$B$3$N6K>.<+M3J,2r$O<B$O(B $B9TNs(B $(1,2,3)$ $B$G$-$^$k(B affine toric ideal |
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$B$N6K>.<+M3J,2r$N(B Koszul complex $B$K$J$C$F$k(B. |
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Gel'fand, Kapranov, Zelevinsky $B$K$h$C$FF3F~$5$l$?(B $D/I$ |
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$B$N(B resolution $B$r<+A3$K1dD9$7$?<!$N(B 2 $B=EJ#BN$r9M$($h$&(B. |
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$$ |
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\begin{array}{ccccccccc} |
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0 & \longrightarrow & D^2 & \stackrel{d^1}{\longrightarrow} |
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& D^3 & \stackrel{d^2}{\longrightarrow} |
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& D & \longrightarrow & 0 \\ |
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& & u^1 \downarrow & |
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& u^2 \downarrow & |
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& u^3 \downarrow & \\ |
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0 & \longrightarrow & D^2 & \stackrel{d^1}{\longrightarrow} |
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& D^3 & \stackrel{d^2}{\longrightarrow} |
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& D & \longrightarrow & 0 |
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\end{array} |
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$$ |
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$B$3$3$G$O(B, ($B8m2r$b$J$$$H;W$&$N$G(B) $D$ $B$GF1<!2=%o%$%kBe?t(B, |
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$d^i$ $B$G(B affine toric ideal $B$NF1<!2=$NB?9`<04D$G$N6K>.<+M3J,2r(B |
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$$ d^2 = \pmatrix{ \pd{1}^2 - \pd{2}^2 h \cr |
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-\pd{1} \pd{2} + \pd{3} h \cr |
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\pd{2}^2 - \pd{1} \pd{3} \cr }, \ |
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d^1 = \pmatrix{ -\pd{2} & -\pd{1} & -h \cr |
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\pd{3} & \pd{2} & \pd{1} \cr } |
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$$ |
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$B$r$"$i$o$9$b$N$H$9$k(B. |
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$B$^$?(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 |
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$B<!$N$h$&$K$-$a$k(B. |
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$$ u^1=\pmatrix{ \ell + 4 h^2 & 0 \cr |
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0 & \ell+5 h^2 \cr}, \quad |
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u^2=\pmatrix{\ell+2 h^2 & 0 & 0 \cr |
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0 & \ell + 3 h^2 & 0 \cr |
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0 & 0 & \ell+ 4 h^2 \cr}, \quad |
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u^3 = \pmatrix{ \ell \cr}. |
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$$ |
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|
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$B$3$N$H$-IU?o$9$k(B 1 $B=EJ#BN$O(B |
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$$L^1 \ni f \mapsto (-d^1(f), u^1(f)) \in L^2 \oplus L^1, $$ |
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$$ L^2\oplus L^1 \ni (f,g)\mapsto (-d^2(f), u^2(f)+d^1(g)) \in L^3\oplus L^2, |
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$$ |
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$$ |
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L^3\oplus L^2 \ni (f,g)\mapsto u^3(f)+d^2(g) \in L^3. |
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$$ |
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$B$G$"$?$($i$l$k(B. |
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$B$3$3$G(B $L^1 = D^2$, $L^2 = D^3$, $L^3 = D$ $B$G$"$k(B. |
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$B$3$N(B 1 $B=EJ#BN$N6qBN7A$O0J2<$N$H$&$j(B. |
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\footnotesize{ |
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\begin{verbatim} |
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[ |
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[ |
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[ x1*Dx1+2*x2*Dx2+3*x3*Dx3 ] |
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[ Dx1^2-Dx2*h ] |
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[ -Dx1*Dx2+Dx3*h ] |
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[ Dx2^2-Dx1*Dx3 ] |
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] |
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[ |
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[ -Dx1^2+Dx2*h , x1*Dx1+2*x2*Dx2+3*x3*Dx3+2*h^2 , 0 , 0 ] |
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[ Dx1*Dx2-Dx3*h , 0 , x1*Dx1+2*x2*Dx2+3*x3*Dx3+3*h^2 , 0 ] |
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[ -Dx2^2+Dx1*Dx3 , 0 , 0 , x1*Dx1+2*x2*Dx2+3*x3*Dx3+4*h^2 ] |
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[ 0 , -Dx2 , -Dx1 , -h ] |
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[ 0 , Dx3 , Dx2 , Dx1 ] |
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] |
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[ |
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[ Dx2 , Dx1 , h , x1*Dx1+2*x2*Dx2+3*x3*Dx3+4*h^2 , 0 ] |
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[ -Dx3 , -Dx2 , -Dx1 , 0 , x1*Dx1+2*x2*Dx2+3*x3*Dx3+5*h^2 ] |
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] |
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] |
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\end{verbatim} |
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} |
| \end{example} |
\end{example} |
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| $(-w,w)$-$B6K>.<+M3J,2r$H(B $B6K>.<+M3J,2r$,$3$H$J$kNc$r$5$,$7$F$$$k$,(B |
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| $B$3$l$O$^$@8+$D$+$C$F$$$J$$(B. |
|
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|
| \section{$B<BAu(B} |
\section{$B<BAu(B} |
| $B$3$3$G$O(B |
$B$3$3$G$O(B |
| \begin{verbatim} |
\begin{verbatim} |
| /* OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v |
/* OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v 1.25 |
| 1.23 2000/08/01 08:51:03 takayama Exp */ |
2000/08/02 05:14:31 takayama Exp */ |
| \end{verbatim} |
\end{verbatim} |
| $BHG$N(B {\tt minimal.k} $B$K=`5r$7$F<BAu$N35N,$r2r@b$9$k(B. |
$BHG$N(B {\tt minimal.k} $B$K=`5r$7$F<BAu$N35N,$r2r@b$9$k(B. |
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| $B$^$@=q$$$F$J$$(B. |
$B<BAu$N@bL@$N$?$a$NNc$H$7$F%$%G%"%k(B |
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$$ I = D \cdot \{ -2x\pd{x}-3y\pd{y}+h^2, -3y\pd{x}^2+2x\pd{y}h \} $$ |
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$B$N(B $(u,v) = (-1,-1,1,1)$-$B6K>.J,2r$N9=@.(B |
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$B$r9M$($h$&(B. |
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%%Prog: minimal-note-ja.txt 6/9 (Fri) $B$*$h$S0J8e$N(B bug fix $B$N5-O?$r;2>H(B. |
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%%$BNc$H$7$F(B, $B%$%G%"%k(B |
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%%$$ I = D \cdot \{ x^2 + y^2, x y \} $$ |
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%%$B$N(B $(u,v) = (-1,-1,1,1)$-$B6K>.J,2r$N9=@.(B |
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%%$B$r9M$($h$&(B. |
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%%($B$3$N>l9g$OB?9`<04D$NF1<!<0$G@8@.$5$l$k$N$G(B, $BB?9`<04D$G$N(B |
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%% $B6K>.<+M3J,2r$N7W;;$HF1$8$3$H$K$J$k(B.) |
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$B$3$N>l9g(B, |
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$I$ $B$N%0%l%V%J4pDl(B $G$ $B$O(B |
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{\footnotesize |
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\begin{verbatim} |
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[ |
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[ -2*x*Dx-3*y*Dy+h^2 ] |
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[ -3*y*Dx^2+2*x*Dy*h ] |
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[ 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ] |
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[ 27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ] |
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] |
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\end{verbatim} |
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} \noindent |
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$B$H$J$C$F$*$j(B, |
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Schreyer resolution $B$O(B |
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{\footnotesize |
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\begin{verbatim} |
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[ |
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[ |
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[ -2*x*Dx-3*y*Dy+h^2 ] |
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[ -3*y*Dx^2+2*x*Dy*h ] |
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[ 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ] |
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[ 27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ] |
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] |
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[ |
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[ 9*y^2*Dy+3*y*h^2 , 0 , 2*x , 1 ] |
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[ -4*x^2*Dy*h , 0 , -3*y*Dy+4*h^2 , Dx ] |
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[ 2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ] |
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[ 3*y*Dx , -2*x , 1 , 0 ] |
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] |
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[ |
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[ -Dx , 1 , 2*x , 3*y*Dy-2*h^2 ] |
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] |
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] |
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\end{verbatim} |
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} \noindent |
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$B$G$"$k(B. $1$ $B$,$?$/$5$s(B Schreyer resolution $B$NCf$K$O$"$k$3$H$K(B |
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$BCm0U(B. $1$ $B$O6K>.<+M3J,2r$K$OI,MW$J$$85$G$"$k$3$H$r0UL#$9$k(B. |
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$B6K>.<+M3J,2r$O(B, $BNc(B \ref{example:cusp} $B$K6qBN7A$r=q$$$F$*$$$?(B. |
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\medbreak |
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|
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$B$3$N<BAu$G$O6K>.<+M3J,2r$r(B LaScala $B$N%"%k%4%j%:%`$r$b$H$K$7$F(B |
| |
$B9=@.$9$k(B (LaScala and Stillman [??] $B$*$h$S?tM}2J3X$N5-;v(B ??? $B$r;2>H(B). |
| |
|
| |
$B$3$N%"%k%4%j%:%`$O4{CN$H$7$F(B, $B0c$$$N$_$r@bL@$7$h$&(B. |
| |
LaScala $B$N%"%k%4%j%:%`$O(B, |
| |
reduction $B$7$?$H$-$K(B $0$ $B$K$J$C$?>l9g(B, $B$=$N(B reduction $B$KIU?o$7$?(B |
| |
syzygy $B$r(B $B6K>.<+M3J,2r$N85$H$7(B, |
| |
reduction $B$7$?$H$-$K(B $0$ $B$K$J$i$J$+$C$?>l9g(B, $B$=$N85$r(B |
| |
$B%0%l%V%J4pDl$N85$H$7$F2C$((B, $BIU?o$7$?(B syzygy $B$O6K>.<+M3J,2r$G$OM>7W$J$b$N$H(B |
| |
$B$_$J$9(B. |
| |
$B$o$l$o$l$O(B $(u,v)$-$B6K>.$J<+M3J,2r$r$b$H$a$?$$(B. |
| |
$B$=$3$G>e$N<jB3$-$r<!$N$h$&$KJQ$($k(B. |
| |
\begin{center} |
| |
\begin{minipage}{10cm} |
| |
Reduction $B$7$?$H$-$K(B modulo $(u,v)$-$B%U%#%k%?!<$G(B $0$ $B$K$J$C$?>l9g(B, |
| |
$B$=$N(B reduction $B$KIU?o$7$?(B syzygy $B$r(B $B6K>.<+M3J,2r$N85$H$7(B, $B$5$i$K(B |
| |
$B$=$N85$,(B $0$ $B$G$J$1$l$P%0%l%V%J4pDl$K2C$($k(B. |
| |
reduction $B$7$?$H$-$K(B modulo $(u,v)$-$B%U%#%k%?!<$G(B $0$ $B$K$J$i$J$+$C$?>l9g(B, |
| |
$B$=$N85$r%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$_$J$9(B. |
| |
\end{minipage} |
| |
\end{center} |
| |
|
| |
$B$A$J$_$K(B, |
| |
$(-w,w)$-$B6K>.<+M3J,2r$H(B $B6K>.<+M3J,2r$,$3$H$J$kNc$r$5$,$7$F$$$k$,(B |
| |
$B$3$l$O$^$@8+$D$+$C$F$$$J$$(B. |
| |
$B$A$g$C$HIT;W5D$G$"$k(B. |
| |
|
| |
\bigbreak |
| |
|
| |
{\tt minimal.k} $B$N%=!<%9%3!<%I$G$O$3$NItJ,$O<!$N$h$&$K$J$C$F$$$k(B. |
| |
{\footnotesize |
| |
\begin{verbatim} |
| |
def SlaScala(g,opt) { |
| |
... |
| |
... |
| |
f = SpairAndReduction(skel,level,i,freeRes,tower,ww); |
| |
if (f[0] != Poly("0")) { |
| |
place = f[3]; |
| |
if (Sordinary) { |
| |
redundantTable[level-1,place] = redundant_seq; |
| |
redundant_seq++; |
| |
}else{ |
| |
if (f[4] > f[5]) { ($B$$(B) |
| |
/* Zero in the gr-module */ |
| |
Print("v-degree of [org,remainder] = "); |
| |
Println([f[4],f[5]]); |
| |
Print("[level,i] = "); Println([level,i]); |
| |
redundantTable[level-1,place] = 0; |
| |
}else{ ($B$m(B) |
| |
redundantTable[level-1,place] = redundant_seq; |
| |
redundant_seq++; |
| |
} |
| |
} |
| |
redundantTable_ordinary[level-1,place] |
| |
=redundant_seq_ordinary; |
| |
... |
| |
... |
| |
} |
| |
\end{verbatim} |
| |
} |
| |
|
| |
$B>/!9D9$/$J$k$,(B, $B$3$NItJ,$K$"$i$o$l$kJQ?t$N@bL@$r$7$h$&(B. |
| |
|
| |
LaScala $B$N%"%k%4%j%:%`$G$O(B, $B:G=i$K7W;;$9$Y$-(B S-pair $B$N7W;;<j=g(B, |
| |
$B$*$h$S(B Schreyer frame $B$r:n@.$9$k(B. |
| |
Schreyer frame $B$O(B Schreyer resolution $B$N(B initial $B$G$"$k(B. |
| |
$B$3$l$i$O$"$i$+$8$a(B |
| |
{\tt SresolutionFrameWithTower(g,opt);} |
| |
$B$G7W;;$5$l$F(B, {\tt tower} $B$*$h$S(B {\tt skel} $B$K3JG<$5$l$F$$$k(B. |
| |
$B$3$l$i$NJQ?t$NCM$O(B, $B4X?t(B {\tt Sminimal()} $B$NLa$jCM$H$7$F(B |
| |
$B8+$k$3$H$,$G$-$k(B. |
| |
$B4X?t(B {\tt Sminimal()} $B$NLa$jCM$,JQ?t(B $a$ $B$K3JG<$5$l$F$$$k$H$9$k$H(B, |
| |
{\tt a[0]} $B$,6K>.<+M3J,2r(B |
| |
{\tt a[3]} $B$,(B Schreyer $B<+M3J,2r(B($B$H$/$K(B {\tt a[3,0]} $B$,(B |
| |
$I$ $B$N%0%l%V%J4pDl(B), |
| |
{\tt a[4]} $B$,(B, |
| |
$B4X?t(B {\tt SlaScala()} $B$N(B |
| |
$BJQ?t(B {\tt [rf[0], tower, skel, rf[3]]} $B$NCM$G$"$k(B. |
| |
$B$7$?$,$C$F(B, {\tt tower} $B$O(B {\tt a[4,1]} $B$K3JG<$5$l$F$$$k(B. |
| |
$I$ $B$N>l9g$N(B {\tt tower} $B$O0J2<$N$H$&$j(B. |
| |
{\footnotesize |
| |
\begin{verbatim} |
| |
In(25)=sm1_pmat(a[4,1]); |
| |
[ |
| |
[ -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ] |
| |
[ -9*y^2*Dy , -3*es^2*y*Dy , -3*es*y*Dy , -3*y*Dx ] |
| |
[ -Dx ] |
| |
] |
| |
\end{verbatim} |
| |
} \noindent |
| |
$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$$$k(B. |
| |
$B$?$H$($P(B, |
| |
\verb# -3*es^2*y*Dy # $B$O(B |
| |
\verb# [0, 0, -3*y*Dy, 0] # $B$r0UL#$9$k(B. |
| |
|
| |
$BJQ?t(B |
| |
{\tt skel} $B$K$O(B |
| |
S-pair (sp) $B$N7W;;<j=g$,$O$$$C$F$$$k(B. |
| |
$I$ $B$N>l9g$K$O0J2<$N$H$&$j(B. |
| |
{\footnotesize |
| |
\begin{verbatim} |
| |
In(16)=sm1_pmat(a[4,2]); |
| |
[ |
| |
[ ] |
| |
[ |
| |
[ |
| |
[ 0 , 2 ] G'[0] $B$H(B G'[2] $B$N(B sp $B$r7W;;(B (0) |
| |
[ -9*y^2*Dy , 2*x ] |
| |
] |
| |
[ |
| |
[ 2 , 3 ] G'[2] $B$H(B G'[3] $B$N(B sp $B$r7W;;(B (1) |
| |
[ -3*y*Dy , Dx ] |
| |
] |
| |
[ |
| |
[ 1 , 2 ] G'[1] $B$H(B G'[2] $B$N(B sp $B$r7W;;(B (2) |
| |
[ -3*y*Dy , Dx ] |
| |
] |
| |
[ |
| |
[ 0 , 1 ] G'[0] $B$H(B G'[1] $B$N(B sp $B$r7W;;(B (3) |
| |
[ -3*y*Dx , 2*x ] |
| |
] |
| |
] |
| |
[ |
| |
[ |
| |
[ 0 , 3 ] G''[0] $B$H(B G''[3] $B$N(B sp $B$r7W;;(B |
| |
[ -Dx , 3*y*Dy ] |
| |
] |
| |
] |
| |
[ ] |
| |
] |
| |
\end{verbatim} |
| |
} \noindent |
| |
$B$3$3$G(B $G'$ $B$O(B Schreyer order $B$GF@$i$l$?(B $G$ $B$N(B syzygy $B$N@8@.85(B, |
| |
$G''$ $B$O(B Schreyer order $B$GF@$i$l$?(B $G'$ $B$N(B syzygy $B$N@8@.85$r$"$i$o$9(B. |
| |
$B$?$H$($P>e$NNc$G$O(B, |
| |
$G'[0]$ $B$O(B |
| |
$G[0]$ $B$H(B $G[2]$ $B$N(B sp $B$N7W;;$K$h$jF@$i$l$?(B syzygy, |
| |
$G'[1]$ $B$O(B |
| |
$G[2]$ $B$H(B $G[3]$ $B$N(B sp $B$N7W;;$K$h$jF@$i$l$?(B syzygy, |
| |
... |
| |
$B$r0UL#$9$k(B. |
| |
|
| |
{\footnotesize |
| |
\begin{verbatim} |
| |
f = SpairAndReduction(skel,level,i,freeRes,tower,ww); |
| |
\end{verbatim} |
| |
} \noindent |
| |
$B$G$O(B {\tt skel[level,i]} $B$K3JG<$5$l$?(B |
| |
S-pair $B$r7W;;$7$F(B, {\tt freeRes[level-1]} $B$G(B reduction $B$r$*$3$J$&(B. |
| |
Reduction $B$N$?$a$N(B Schreyer order $B$O(B \\ |
| |
{\tt StowerOf(tower,level-1)} $B$rMQ$$$k(B. |
| |
$B$?$H$($P(B, ${\tt [level,i] = [1,3]}$ $B$N$H$-$K(B |
| |
$B4X?t(B {\tt SpairAndReduction} $B$G(B |
| |
$B$I$N$h$&$J7W;;$,$J$5$l$F$$$k$+(B $I$ $B$N>l9g$K$_$F$_$h$&(B. |
| |
|
| |
{\tt SpairAndReduction} $B$N<B9T;~(B |
| |
$B$K<!$N$h$&$J%a%C%;!<%8$,$G$F$/$k(B. |
| |
{\footnotesize |
| |
\begin{verbatim} |
| |
reductionTable= [ |
| |
[ 1 , 2 , 3 , 4 ] |
| |
[ 3 , 4 , 3 , 2 ] |
| |
[ 3 ] |
| |
] |
| |
[ 0 , 0 ] |
| |
Processing [level,i]= [ 0 , 0 ] Strategy = 1 |
| |
[ 0 , 1 ] |
| |
Processing [level,i]= [ 0 , 1 ] Strategy = 2 |
| |
[ 1 , 3 ] |
| |
Processing [level,i]= [ 1 , 3 ] Strategy = 2 |
| |
SpairAndReduction: |
| |
[ p and bases , [ [ 0 , 1 ] , [ -3*y*Dx , 2*x ] ] , |
| |
[-2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ] ] |
| |
[ level= , 1 ] |
| |
[ tower2= , [ [ ] ] ] |
| |
[ -3*y*Dx , 2*es*x ] |
| |
[gi, gj] = [ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h ] |
| |
1 |
| |
Reduce the element 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h |
| |
by [ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ] |
| |
result is [ 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , 1 , [ 0 , 0 , 0 , 0 ] ] |
| |
vdegree of the original = 0 |
| |
vdegree of the remainder = 0 |
| |
[ 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , |
| |
[ -3*y*Dx , 2*x , 0 , 0 ] , 3 , 2 , 0 , 0 ] |
| |
\end{verbatim} |
| |
} \noindent |
| |
$B:G=i$KI=<($5$l$k(B {\tt reductionTable} $B$N0UL#$O$"$H$G@bL@$9$k(B. |
| |
$B<!$N9T$KCmL\$7$h$&(B. $B$3$3$G$O(B {\tt skel[0,4]} $B$N(B S-pair |
| |
$B$r7W;;$7$F(Breduction $B$7$F$$$k(B. |
| |
{\footnotesize |
| |
\begin{verbatim} |
| |
SpairAndReduction: |
| |
[ p and bases , [ [ 0 , 1 ] , [ -3*y*Dx , 2*x ] ] , |
| |
[-2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ] ] |
| |
\end{verbatim} |
| |
} \noindent |
| |
{\tt [0, 1]} $B$O(B $G'[0]$ $B$H(B $G'[1]$ $B$N(B sp $B$r7W;;(B |
| |
$B$;$h$H$$$&0UL#$G$"$k(B. |
| |
${\tt level} = 0$ $B$G4{$K$b$H$^$C$F$$$k(B $B%V%l%V%J4pDl$O(B |
| |
$G[0]$ $B$H(B $G[1]$ $B$N$_$G$"$j(B, |
| |
$B$=$l$i$O$=$l$>$l(B, |
| |
\verb# -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h # |
| |
$B$G$"$k(B. |
| |
{\tt SpairAndReduction} $B$O(B $G[0]$, $G[1]$ $B$N$_$rMQ$$$F(B, |
| |
S-pair \\ |
| |
\verb# 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h # |
| |
$B$r(B reduction $B$9$k(B. |
| |
$B7k6I(B reduction $B$N7k2L$O(B 0 $B$G$O$J$/$F(B, \\ |
| |
\verb# 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h # |
| |
$B$H$J$k(B. |
| |
LaScala $B$N%"%k%4%j%:%`$N(B 2 $BG\$*F@%7%9%F%`$G(B, |
| |
$B$3$l$,?7$7$$%0%l%V%J4pDl$N85(B {\tt G[place]} $B$H$J$j(B, |
| |
reduction $B$N2aDx$h$j(B syzygy $B$bF@$i$l$k(B. |
| |
|
| |
$B$5$F(B, $(u,v)$-$B6K>.J,2r$r:n$k$K$O(B, reduction $B$7$?M>$j$,(B |
| |
$(u,v)$-$B%U%#%k%?!<$G(B modulo $B$7$F(B $0$ $B$+$I$&$+D4$Y$J$$$H$$$1$J$$(B. |
| |
$B$3$N$?$a(B, |
| |
$B4X?t(B {\tt Sdegree()} $B$rMQ$$$F(B, reduction $B$9$kA0$N85(B, $B$*$h$SM>$j$N(B |
| |
$B%7%U%HIU$-(B $(u,v)$-order $B$r7W;;$9$k(B. |
| |
$B$3$NNc$G$O(B, $BN>J}$H$b(B $0$ $B$G$"$k(B. |
| |
{\footnotesize |
| |
\begin{verbatim} |
| |
vdegree of the original = 0 |
| |
vdegree of the remainder = 0 |
| |
\end{verbatim} |
| |
} |
| |
$B$7$?$,$C$F(B, modulo $(u,v)$-$B%U%#%k%?!<$G$b(B $0$ $B$G$J$$(B. |
| |
|
| |
$B=`Hw@bL@$,$*$o$C$?(B. $B:G=i$N%W%m%0%i%`(B {\tt SlaScala()} $B$N@bL@$KLa$k(B. |
| |
{\tt SpairAndReduction()} $B$NLa$jCM(B |
| |
{\tt f[0]} $B$K$O(B, reduction $B$7$?M>$j(B, |
| |
{\tt f[4]}, {\tt f[5]} $B$K$O(B, |
| |
reduction $B$9$kA0$N85(B, $B$*$h$SM>$j$N(B |
| |
$B%7%U%HIU$-(B $(u,v)$-order $B$,3JG<$5$l$F$$$k(B. |
| |
$B$3$NNc$N>l9g$K$O(B ($B$m(B) $B$N>l9g$,<B9T$5$l$F(B, |
| |
$BIU?o$7$?(B syzygy $B$O(B $B6K>.<+M3J,2r$K$OITMW$J$b$N$H$7$F(B |
| |
{\tt redundantTable} $B$KEPO?$5$l$k(B: |
| |
{\footnotesize |
| |
\begin{verbatim} |
| |
redundantTable[level-1,place] = redundant_seq; |
| |
\end{verbatim} |
| |
} \noindent |
| |
$BM>$j(B {\tt f[0]} $B$O(B, laScala $B$N%"%k%4%j%:%`$N(B 2 $BG\$*F@8=>]$GF@$i$l$?(B, |
| |
$B?7$7$$%V%l%V%J4pDl$N85$G$"$k$,(B, $B$3$l$rJ]B8$9$Y$->l=j$N%$%s%G%C%/%9$O(B, |
| |
$BLa$jCM(B {\tt f[3]}({\tt place}) $B$K3JG<$5$l$F$$$k(B: |
| |
{\footnotesize |
| |
\begin{verbatim} |
| |
bases[place] = f[0]; |
| |
freeRes[level-1] = bases; |
| |
reducer[level-1,place] = f[1]; |
| |
\end{verbatim} |
| |
} \noindent |
| |
$B$3$N(B reduction $B$GF@$i$l$?(B syzygy ($B$NK\<AE*ItJ,(B)$B$O(B, |
| |
$BJQ?t(B {\tt reducer} $B$KEPO?$5$l$k(B. |
| |
$B0J>e$G(B $(u,v)$-$B6K>.<+M3J,2rFCM-$N=hM}$NItJ,$N2r@b$r=*$($k(B. |
| |
|
| |
|
| |
\bigbreak |
| |
$B0J2<$G$O(B, LaScala $B$N%"%k%4%j%:%`$N$o$l$o$l$N<BAu$N35N,$HLdBjE@$r(B |
| |
$B=R$Y$k(B. |
| |
|
| |
$B$^$:(B, $BJQ?t(B |
| |
{\tt reductionTable} $B$N0UL#$r@bL@$7$h$&(B. |
| |
LaScala $B$N%"%k%4%j%:%`$G$O(B, |
| |
{\tt level - Sdegree(s)} |
| |
$B$N>.$5$$(B S-pair $B$+$i7W;;$7$F$$$/(B. |
| |
$B4X?t(B {\tt Sdegree} $B$O<!$N$h$&$K:F5"E*$KDj5A$5$l$F$$$k(B. |
| |
{\footnotesize |
| |
\begin{verbatim} |
| |
/* f is assumed to be a monomial with toes. */ |
| |
def Sdegree(f,tower,level) { |
| |
local i,ww, wd; |
| |
/* extern WeightOfSweyl; */ |
| |
ww = WeightOfSweyl; |
| |
f = Init(f); |
| |
if (level <= 1) return(StotalDegree(f)); |
| |
i = Degree(f,es); |
| |
return(StotalDegree(f)+Sdegree(tower[level-2,i],tower,level-1)); |
| |
} |
| |
\end{verbatim} |
| |
} \noindent |
| |
$B$3$3$G(B {\tt StotalDegree(f)} $B$O(B $f$ $B$NA4<!?t$G$"$k(B. |
| |
|
| |
\noindent |
| |
$B$5$F(B, LaScala $B$N%"%k%4%j%:%`$G$O(B, |
| |
Resolution $B$r2<$+$i=gHV$K7W;;$7$F$$$/$N$G$O$J$$(B. |
| |
$B$3$l$,K\<AE*$JE@$G$"$k(B. |
| |
$B$3$N=gHV$OJQ?t(B {\tt reductionTable} $B$K$O$C$F$$$k(B. |
| |
$I$ $B$NNc$G$O(B |
| |
{\footnotesize |
| |
\begin{verbatim} |
| |
reductionTable= [ |
| |
[ 1 , 2 , 3 , 4 ] |
| |
[ 3 , 4 , 3 , 2 ] skel[0] $B$KBP1~(B |
| |
[ 3 ] skel[1] $B$KBP1~(B |
| |
] |
| |
\end{verbatim} |
| |
} \noindent |
| |
$B$H$J$k(B. |
| |
|
| |
$B8=:_$N<BAu$G$N7W;;B.EY(B, $B%a%b%j;HMQNL$N%\%H%k%M%C%/$r(B |
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$B;XE&$7$F$*$/(B. |
| |
LaScala $B$N%"%k%4%j%:%`$G$O(B, Schreyer Frame $B$r9=@.$7$F$+$i(B, |
| |
$B6K>.<+M3J,2r$r9=@.$9$k(B. |
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$B2<5-$N%W%m%0%i%`$NJQ?t(B {\tt redundantTable[level,q]} $B$K$O(B, |
| |
$BBP1~$9$k(B syzygy $B$H(B $B%0%l%V%J4pDl$N85$,2?2sL\$N(B reduction $B$G@8@.(B |
| |
$B$5$l$?$+$N?t$,$O$$$C$F$$$k(B. |
| |
$B6K>.<+M3J,2r$N9=@.$G$O(B, $B:G8e$N(B reduction $B$N(B syzygy $B$+$i;O$a$F(B, |
| |
Schreyer resolution $B$+$i6K>.<+M3J,2r$K$H$C$FM>J,$J85$r<h$j=|$$$F(B |
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$B$$$/(B |
| |
({\tt seq} $B$r(B $1$ $B$E$D8:$i$7$F$$$/(B). |
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{\footnotesize |
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\begin{verbatim} |
| |
def Sminimal(g,opt) { |
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|
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.... |
| |
|
| |
while (seq > 1) { |
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seq--; |
| |
for (level = 0; level < maxLevel; level++) { |
| |
betti = Length(freeRes[level]); |
| |
for (q = 0; q<betti; q++) { |
| |
if (redundantTable[level,q] == seq) { |
| |
Print("[seq,level,q]="); Println([seq,level,q]); |
| |
if (level < maxLevel-1) { |
| |
bases = freeRes[level+1]; |
| |
dr = reducer[level,q]; |
| |
dr[q] = -1; |
| |
newbases = SnewArrayOfFormat(bases); |
| |
betti_levelplus = Length(bases); |
| |
/* |
| |
bases[i,j] ---> bases[i,j]+bases[i,q]*dr[j] |
| |
*/ |
| |
for (i=0; i<betti_levelplus; i++) { |
| |
newbases[i] = bases[i] + bases[i,q]*dr; |
| |
} |
| |
.... |
| |
} |
| |
.... |
| |
} |
| |
} |
| |
} |
| |
} |
| |
.... |
| |
} |
| |
\end{verbatim} |
| |
} \noindent |
| |
$BLdBj$O(B, |
| |
$B6K>.<+M3J,2r<+BN$O$A$$$5$/$F$b(B, Schreyer Frame $B$,5pBg(B ($10000$ $BDxEY$N(B |
| |
betti $B?t(B) $B$H$J$k$3$H$bB?$$>l9g$,$"$k$3$H$G$"$k(B. |
| |
$B2<$NJQ?t(B {\tt bases} $B$K(B, Schreyer resolution $B$N(B {\tt level} $B<!$N(B |
| |
syzygy $B$r$$$l$F$$$k(B. Schreyer Frame $B$K(B $10000$ $BDxEY$N(B betti |
| |
$B?t$,$"$i$o$l$k$H$3$NJQ?t$O(B $B%5%$%:(B $10000$ $BDxEY$NG[Ns$H$J$k(B. |
| |
$B$5$i$K(B, Schreyer $BJ,2r$+$i6K>.<+M3J,2r$N$?$a$KITMW$J85$r$H$j$N$>$$$?(B |
| |
$BJ,2r$r:n$k$?$a$K(B\\ |
| |
\verb# newbases[i] = bases[i] + bases[i,q]*dr; # \\ |
| |
$B$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. |
| |
$B$3$NItJ,$,(B, $B%a%b%j$N;HMQ$r05Gw$7$F$*$j(B, $B7W;;;~4V$b$D$+$C$F$$$k(B. |
| |
|
| |
|
| |
|
| \end{document} |
\end{document} |
| |
|
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