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I added some codes to check if the output is minimal or not.

$OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.1 2000/05/19 11:16:51 takayama Exp $   

SpairAndReduction() :
   $BM?$($i$l$?(B pair $B$r(B reduction $B$9$k(B.
   V-minimal $B$KI,MW$+$I$&$+$NH=Dj$b$9$k(B.

SpairAndReduction2():
  tower2 = StowerOf(tower,level-1);
  SsetTower(tower2);  
  /** sm1(" show_ring ");   */

  $BM?$($i$l$?(B pair $B$r(B reduction $B$9$k$?$a$N(B schreyer order
  $B$r@_Dj$9$k(B.  Resolution $B$N?<$5$K1~$8$F(B, tower $B$b?<$/$9$kI,MW$,$"$k(B.


  if (IsConstant(t_syz[i])){

  Syzygy $B$r$_$F(B, $BDj?t@.J,$,$J$$$+(B check.
  t_syz[i] $B$,Dj?t@.J,$G$"$l$P(B, $B0l$DA0$N(B GB $B$N9=@.MWAG$G$"$k(B
  g_i $B$,M>J,$J(B GB $B$G$"$k2DG=@-$,$?$+$$(B.
  SpairAndReduction() ( LaScala-Stillman $B$NJ}K!(B) $B$H$N@09g@-$r$H$k$?$a(B 
  g_i $B$r(B tmp[0] $B$KBeF~$7(B ( reduction $B$G$-$J$+$C$?$U$j$r$9$k(B )
  g_i $B$N(B V-degree $B$r$7$i$Y$k(B.


Sannfs2_laScala2()
Sannfs3_laScala2()  $B$r:n$k(B.

$BFs$D$N%"%k%4%j%:%`$NHf3S(B.
In(11)=sm1_pmat(a1[1]); $B$N=gHV$r$+$($k(B.
 [ 
   [    3*Dx^2*h , 0 , Dy , -Dz ] 
   [    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]
   [    2*x*Dx+3*z*Dz-h^2 , y*Dy-z*Dz , 0 , 0 ] 
   [    2*x*Dy*Dz , 0 , z , -y ] 

   [    0 , 2*x*Dy^2*Dz-3*z*Dx^2*h , 0 , 2*x*Dx+3*z*Dz ] 
 ]
In(12)=sm1_pmat(a2[1]);
 [ 
   [    3*Dx^2*h , 0 , Dy , -Dz ] 
   [    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 ] 
   [    2*x*Dx+3*z*Dz-h^2 , y*Dy-z*Dz , 0 , 0 ] 
   [    2*x*Dy*Dz , 0 , z , -y ] 

   [    9*z*Dx^2*h , 2*x*Dy^2*Dz-3*z*Dx^2*h , 3*z*Dy , 2*x*Dx ] 
   [    2*x*Dx*Dz^2+3*z*Dz^3+5*Dz^2*h^2 , y*Dy*Dz^2-z*Dz^3-2*Dz^2*h^2 , 0 , 0 ] 
 ]
In(13)=

----------------------
In(16)=sm1_pmat(a1[2]);
 [ 
   [    -2*x*Dx-3*y*Dy-3*z*Dz-6*h^2 , -Dy , -Dz , 3*Dx^2*h , 3*Dy^2 , 3*Dy*Dz , -2*x*Dy , 2*x*Dz , 0 ] 
   [    3*y*z , z , y , -2*x*Dy*Dz , -3*z*Dy , 2*x*Dx , 2*x*z , -2*x*y , 0 ] 
 ]
In(17)=sm1_pmat(a2[2]);
 [ 
   [    -y , 2*x*Dy*Dz , z , 0 , 2*x*Dx , 0 ] 
   [    -Dz , 3*Dx^2*h , Dy , -2*x*Dx-3*y*Dy-3*h^2 , -3*Dy*Dz , 0 ] 
 ]
In(18)=